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Номер 26, страница 72, часть 2 - гдз по алгебре 10 класс учебник Пак, Ардакулы
Авторы: Пак О. В., Ардакулы Д., Ескендирова Е. В.
Тип: Учебник
Издательство: Алматыкітап баспасы
Год издания: 2019 - 2026
Часть: 2
ISBN: 978-601-01-3958-9
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 10 классе
Часть 2. Глава 5. Производная. Параграф 3. Физический и геометрический смысл производной. 3.2. Касательная к графику функции. Задачи - номер 26, страница 72.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1120284 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "72" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/5-3-2zad-26" "field_display_title" => "26" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1072 #items: array:1 [ 0 => App\Models\Branch {#1034 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:1 [ 0 => App\Models\Term {#1038 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #original: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1040 #items: array:1 [ 0 => App\Models\Book {#1041 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4295 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1042 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4295 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1042 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1040} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1069} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1143 #items: array:5 [ 0 => App\Models\Branch {#1155 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1159 #items: array:1 [ 0 => App\Models\Term {#1156 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #original: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1188 #items: array:1 [ 0 => App\Models\Book {#1158 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4295 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1160 #items: array:1 [ 0 => App\Models\Term {#1157 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1161 #items: array:1 [ 0 => App\Models\Term {#1162 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1163 #items: array:1 [ 0 => App\Models\Term {#1164 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 7006 "created_at" => "2026-04-10 13:58:26" …4 ] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1165 #items: array:3 [ 0 => App\Models\Term {#1166 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#1167 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Term {#1168 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1169 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1170 #items: array:1 [ 0 => App\Models\Term {#1171 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1172 #items: array:1 [ 0 => App\Models\Term {#1173 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1174 #items: array:1 [ 0 => App\Models\Term {#1175 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1176 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1177 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1178 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1179 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1180 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1181 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1182 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1183 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1184 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1185 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #original: array:50 [ "id" => 4295 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1160} "field_class" => Illuminate\Database\Eloquent\Collection {#1161} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1163} "field_author" => Illuminate\Database\Eloquent\Collection {#1165} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1169} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1170} "field_country" => Illuminate\Database\Eloquent\Collection {#1172} "field_city" => Illuminate\Database\Eloquent\Collection {#1174} "field_series" => Illuminate\Database\Eloquent\Collection {#1176} "field_umk" => Illuminate\Database\Eloquent\Collection {#1177} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1178} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1179} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1180} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1181} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1182} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1183} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1184} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1185} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1186 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1159} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1188} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1186} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1187 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119427 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Производная" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1192 #items: array:1 [ 0 => App\Models\Term {#1189 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1191 #items: array:1 [ 0 => App\Models\Book {#1158} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1190 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119427 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Производная" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1192} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1191} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1190} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Branch {#1193 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119437 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Физический и геометрический смысл производной" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1197 #items: array:1 [ 0 => App\Models\Term {#1194 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "59" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1196 #items: array:1 [ 0 => App\Models\Book {#1158} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1237 #items: array:1 [ 0 => App\Models\Branch {#1195 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119427 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Производная" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1200 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1229 …2} ] #original: array:24 [ "id" => 1119427 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Производная" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1200 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1229 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119437 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Физический и геометрический смысл производной" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1197} "field_page_start" => "59" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1196} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1237} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Branch {#1235 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119439 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3.2. Касательная к графику функции" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1236 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "64" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1238 #items: array:1 [ 0 => App\Models\Book {#1158} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1247 #items: array:1 [ 0 => App\Models\Branch {#1246 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119437 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Физический и геометрический смысл производной" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1248 …2} "field_page_start" => "59" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1249 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1250 …2} ] #original: array:24 [ "id" => 1119437 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Физический и геометрический смысл производной" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1248 …2} "field_page_start" => "59" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1249 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1250 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119439 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3.2. Касательная к графику функции" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1236} "field_page_start" => "64" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1238} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1247} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Branch {#1240 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119441 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1239 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "70" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1243 #items: array:1 [ 0 => App\Models\Book {#1158} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1256 #items: array:1 [ 0 => App\Models\Branch {#1255 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119439 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3.2. Касательная к графику функции" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1257 …2} "field_page_start" => "64" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1258 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1259 …2} ] #original: array:24 [ "id" => 1119439 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3.2. Касательная к графику функции" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1257 …2} "field_page_start" => "64" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1258 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1259 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119441 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1239} "field_page_start" => "70" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1243} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1256} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1104 #items: array:2 [ 0 => App\Models\Element {#1132 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1300969 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1122 #items: array:1 [ 0 => App\Models\Edition {#1131 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4962 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4962 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1120284" "type" => "task" ] "text" => "<p>26.</p><p><strong>(2)</strong> $y = \frac{x^3}{3} + x^2 - x$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "26-1.jpg" "alt" => null "width" => "475" "height" => 185 "path" => "/media/algebra_10/baspasy-u19/0-5-3-2zad/26-1.webp?ts=1753534865" ] ] ] #original: array:7 [ "id" => 1300969 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1122} "task" => array:2 [ "refs" => "1120284" "type" => "task" ] "text" => "<p>26.</p><p><strong>(2)</strong> $y = \frac{x^3}{3} + x^2 - x$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "26-1.jpg" "alt" => null "width" => "475" "height" => 185 "path" => "/media/algebra_10/baspasy-u19/0-5-3-2zad/26-1.webp?ts=1753534865" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1124 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1552005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\Edition {#1125 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5659 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1120 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1115 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5659 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1120 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1115 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1120284" "type" => "task" ] "text" => "<p>Проведем полное исследование функции $y = \frac{x^3}{3} + x^2 - x$ и построим ее график.</p><p><strong>1. Область определения</strong></p><p>Функция является многочленом, поэтому она определена для всех действительных чисел.<br>Ответ: Область определения функции $D(y) = (-\infty, +\infty)$.</p><p><strong>2. Точки пересечения с осями координат</strong></p><p> <em>С осью Oy:</em><br> Для нахождения точки пересечения с осью ординат, подставим $x=0$ в уравнение функции:<br> $y(0) = \frac{0^3}{3} + 0^2 - 0 = 0$.<br> Точка пересечения с осью Oy: $(0, 0)$.</p><p> <em>С осью Ox:</em><br> Для нахождения точек пересечения с осью абсцисс, решим уравнение $y=0$:<br> $\frac{x^3}{3} + x^2 - x = 0$<br> Вынесем $x$ за скобки:<br> $x(\frac{x^2}{3} + x - 1) = 0$<br> Отсюда получаем первый корень $x_1 = 0$.<br> Решим квадратное уравнение $\frac{x^2}{3} + x - 1 = 0$. Умножим обе части на 3:<br> $x^2 + 3x - 3 = 0$<br> Найдем дискриминант: $D = b^2 - 4ac = 3^2 - 4 \cdot 1 \cdot (-3) = 9 + 12 = 21$.<br> Корни уравнения: $x_{2,3} = \frac{-3 \pm \sqrt{21}}{2}$.<br> $x_2 = \frac{-3 - \sqrt{21}}{2} \approx -3.79$, $x_3 = \frac{-3 + \sqrt{21}}{2} \approx 0.79$. <br> Ответ: Точка пересечения с осью Oy: $(0, 0)$. Точки пересечения с осью Ox: $(0, 0)$, $(\frac{-3 - \sqrt{21}}{2}, 0)$, $(\frac{-3 + \sqrt{21}}{2}, 0)$.</p><p><strong>3. Четность и периодичность</strong></p><p> Проверим функцию на четность: $y(-x) = \frac{(-x)^3}{3} + (-x)^2 - (-x) = -\frac{x^3}{3} + x^2 + x$.<br> Так как $y(-x) \neq y(x)$ и $y(-x) \neq -y(x)$, функция не является ни четной, ни нечетной. Это функция общего вида.<br> Функция не является периодической, так как является многочленом. <br> Ответ: Функция общего вида, непериодическая.</p><p><strong>4. Асимптоты графика</strong></p><p> <em>Вертикальные асимптоты:</em><br> Вертикальных асимптот нет, так как функция непрерывна на всей числовой оси.<br> <em>Горизонтальные асимптоты:</em><br> Найдем пределы функции при $x \to \pm\infty$:<br> $\lim_{x \to +\infty} (\frac{x^3}{3} + x^2 - x) = +\infty$<br> $\lim_{x \to -\infty} (\frac{x^3}{3} + x^2 - x) = -\infty$<br> Так как пределы бесконечны, горизонтальных асимптот нет.<br> <em>Наклонные асимптоты:</em><br> Наклонные асимптоты ищутся в виде $y = kx+b$.<br> $k = \lim_{x \to \infty} \frac{y(x)}{x} = \lim_{x \to \infty} \frac{\frac{x^3}{3} + x^2 - x}{x} = \lim_{x \to \infty} (\frac{x^2}{3} + x - 1) = \infty$.<br> Так как предел для $k$ равен бесконечности, наклонных асимптот нет. <br> Ответ: Асимптот нет.</p><p><strong>5. Интервалы монотонности и точки экстремума</strong></p><p> Найдем первую производную функции: <br> $y' = (\frac{x^3}{3} + x^2 - x)' = x^2 + 2x - 1$.<br> Найдем критические точки, приравняв производную к нулю: $y' = 0$.<br> $x^2 + 2x - 1 = 0$<br> $D = 2^2 - 4 \cdot 1 \cdot (-1) = 8$<br> $x_{1,2} = \frac{-2 \pm \sqrt{8}}{2} = \frac{-2 \pm 2\sqrt{2}}{2} = -1 \pm \sqrt{2}$.<br> Критические точки: $x_1 = -1 - \sqrt{2} \approx -2.41$ и $x_2 = -1 + \sqrt{2} \approx 0.41$.<br> Определим знаки производной на интервалах, образованных критическими точками:<br> - На интервале $(-\infty, -1-\sqrt{2})$: $y' > 0$, функция возрастает.<br> - На интервале $(-1-\sqrt{2}, -1+\sqrt{2})$: $y' < 0$, функция убывает.<br> - На интервале $(-1+\sqrt{2}, +\infty)$: $y' > 0$, функция возрастает.<br> В точке $x = -1 - \sqrt{2}$ производная меняет знак с `+` на `-`, следовательно, это точка локального максимума.<br> В точке $x = -1 + \sqrt{2}$ производная меняет знак с `-` на `+`, следовательно, это точка локального минимума.<br> Найдем значения функции в точках экстремума:<br> $y_{max} = y(-1-\sqrt{2}) = \frac{(-1-\sqrt{2})^3}{3} + (-1-\sqrt{2})^2 - (-1-\sqrt{2}) = \frac{5+4\sqrt{2}}{3} \approx 3.55$.<br> $y_{min} = y(-1+\sqrt{2}) = \frac{(-1+\sqrt{2})^3}{3} + (-1+\sqrt{2})^2 - (-1+\sqrt{2}) = \frac{5-4\sqrt{2}}{3} \approx -0.22$.<br> <br> Ответ: Функция возрастает на промежутках $(-\infty, -1-\sqrt{2}]$ и $[-1+\sqrt{2}, +\infty)$, убывает на промежутке $[-1-\sqrt{2}, -1+\sqrt{2}]$. Точка максимума: $(-1-\sqrt{2}, \frac{5+4\sqrt{2}}{3})$. Точка минимума: $(-1+\sqrt{2}, \frac{5-4\sqrt{2}}{3})$.</p><p><strong>6. Интервалы выпуклости/вогнутости и точки перегиба</strong></p><p> Найдем вторую производную функции:<br> $y'' = (x^2 + 2x - 1)' = 2x + 2$.<br> Найдем точки, в которых вторая производная равна нулю: $y''=0$.<br> $2x + 2 = 0 \implies x = -1$.<br> Определим знаки второй производной:<br> - При $x < -1$: $y'' < 0$, график функции выпуклый вверх (вогнутый).<br> - При $x > -1$: $y'' > 0$, график функции выпуклый вниз (выпуклый).<br> Так как в точке $x = -1$ вторая производная меняет знак, это точка перегиба.<br> Найдем значение функции в точке перегиба:<br> $y(-1) = \frac{(-1)^3}{3} + (-1)^2 - (-1) = -\frac{1}{3} + 1 + 1 = \frac{5}{3} \approx 1.67$.<br> <br> Ответ: График функции выпуклый вверх на промежутке $(-\infty, -1]$ и выпуклый вниз на промежутке $[-1, +\infty)$. Точка перегиба имеет координаты $(-1, \frac{5}{3})$.</p><p><strong>7. Построение графика</strong></p><p> Сведем полученные данные в таблицу:</p><table border="1" cellpadding="5" style="border-collapse: collapse; width: 100%;"> <tr> <td style="text-align: center;">$x$</td> <td style="text-align: center;">$(-\infty, \frac{-3-\sqrt{21}}{2})$</td> <td style="text-align: center;">$\frac{-3-\sqrt{21}}{2} \approx -3.79$</td> <td style="text-align: center;">$(\frac{-3-\sqrt{21}}{2}, -1-\sqrt{2})$</td> <td style="text-align: center;">$-1-\sqrt{2} \approx -2.41$</td> <td style="text-align: center;">$(-1-\sqrt{2}, -1)$</td> <td style="text-align: center;">$-1$</td> <td style="text-align: center;">$(-1, 0)$</td> <td style="text-align: center;">$0$</td> <td style="text-align: center;">$(0, -1+\sqrt{2})$</td> <td style="text-align: center;">$-1+\sqrt{2} \approx 0.41$</td> <td style="text-align: center;">$(-1+\sqrt{2}, \frac{-3+\sqrt{21}}{2})$</td> <td style="text-align: center;">$\frac{-3+\sqrt{21}}{2} \approx 0.79$</td> <td style="text-align: center;">$(\frac{-3+\sqrt{21}}{2}, +\infty)$</td> </tr> <tr> <td style="text-align: center;">$y'$</td> <td style="text-align: center;">+</td> <td style="text-align: center;">+</td> <td style="text-align: center;">+</td> <td style="text-align: center;">0</td> <td style="text-align: center;">-</td> <td style="text-align: center;">-</td> <td style="text-align: center;">-</td> <td style="text-align: center;">-</td> <td style="text-align: center;">-</td> <td style="text-align: center;">0</td> <td style="text-align: center;">+</td> <td style="text-align: center;">+</td> <td style="text-align: center;">+</td> </tr> <tr> <td style="text-align: center;">$y''$</td> <td style="text-align: center;">-</td> <td style="text-align: center;">-</td> <td style="text-align: center;">-</td> <td style="text-align: center;">-</td> <td style="text-align: center;">-</td> <td style="text-align: center;">0</td> <td style="text-align: center;">+</td> <td style="text-align: center;">+</td> <td style="text-align: center;">+</td> <td style="text-align: center;">+</td> <td style="text-align: center;">+</td> <td style="text-align: center;">+</td> <td style="text-align: center;">+</td> </tr> <tr> <td style="text-align: center;">$y$</td> <td style="text-align: center;">возрастает, выпукла вверх</td> <td style="text-align: center;">0 (пересечение с Ox)</td> <td style="text-align: center;">возрастает, выпукла вверх</td> <td style="text-align: center;">$\frac{5+4\sqrt{2}}{3}$ (max)</td> <td style="text-align: center;">убывает, выпукла вверх</td> <td style="text-align: center;">$\frac{5}{3}$ (перегиб)</td> <td style="text-align: center;">убывает, выпукла вниз</td> <td style="text-align: center;">0 (пересечение с осями)</td> <td style="text-align: center;">убывает, выпукла вниз</td> <td style="text-align: center;">$\frac{5-4\sqrt{2}}{3}$ (min)</td> <td style="text-align: center;">возрастает, выпукла вниз</td> <td style="text-align: center;">0 (пересечение с Ox)</td> <td style="text-align: center;">возрастает, выпукла вниз</td> </tr></table><p> Основываясь на проведенном исследовании, можно построить график функции. График представляет собой кубическую параболу, которая начинается в $-\infty$, возрастает до точки максимума $(-2.41, 3.55)$, затем убывает, проходя через точку перегиба $(-1, 1.67)$ и точку начала координат $(0,0)$, достигает минимума в точке $(0.41, -0.22)$, после чего снова возрастает и уходит в $+\infty$. График пересекает ось абсцисс в трех точках: $x \approx -3.79$, $x=0$ и $x \approx 0.79$. <br> Ответ: Полное исследование функции проведено, ключевые точки и интервалы поведения функции найдены, что позволяет построить ее график.</p>" ] #original: array:6 [ "id" => 1552005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116} "task" => array:2 [ "refs" => "1120284" "type" => "task" ] "text" => "<p>Проведем полное исследование функции $y = \frac{x^3}{3} + x^2 - x$ и построим ее график.</p><p><strong>1. Область определения</strong></p><p>Функция является многочленом, поэтому она определена для всех действительных чисел.<br>Ответ: Область определения функции $D(y) = (-\infty, +\infty)$.</p><p><strong>2. Точки пересечения с осями координат</strong></p><p> <em>С осью Oy:</em><br> Для нахождения точки пересечения с осью ординат, подставим $x=0$ в уравнение функции:<br> $y(0) = \frac{0^3}{3} + 0^2 - 0 = 0$.<br> Точка пересечения с осью Oy: $(0, 0)$.</p><p> <em>С осью Ox:</em><br> Для нахождения точек пересечения с осью абсцисс, решим уравнение $y=0$:<br> $\frac{x^3}{3} + x^2 - x = 0$<br> Вынесем $x$ за скобки:<br> $x(\frac{x^2}{3} + x - 1) = 0$<br> Отсюда получаем первый корень $x_1 = 0$.<br> Решим квадратное уравнение $\frac{x^2}{3} + x - 1 = 0$. Умножим обе части на 3:<br> $x^2 + 3x - 3 = 0$<br> Найдем дискриминант: $D = b^2 - 4ac = 3^2 - 4 \cdot 1 \cdot (-3) = 9 + 12 = 21$.<br> Корни уравнения: $x_{2,3} = \frac{-3 \pm \sqrt{21}}{2}$.<br> $x_2 = \frac{-3 - \sqrt{21}}{2} \approx -3.79$, $x_3 = \frac{-3 + \sqrt{21}}{2} \approx 0.79$. <br> Ответ: Точка пересечения с осью Oy: $(0, 0)$. Точки пересечения с осью Ox: $(0, 0)$, $(\frac{-3 - \sqrt{21}}{2}, 0)$, $(\frac{-3 + \sqrt{21}}{2}, 0)$.</p><p><strong>3. Четность и периодичность</strong></p><p> Проверим функцию на четность: $y(-x) = \frac{(-x)^3}{3} + (-x)^2 - (-x) = -\frac{x^3}{3} + x^2 + x$.<br> Так как $y(-x) \neq y(x)$ и $y(-x) \neq -y(x)$, функция не является ни четной, ни нечетной. Это функция общего вида.<br> Функция не является периодической, так как является многочленом. <br> Ответ: Функция общего вида, непериодическая.</p><p><strong>4. Асимптоты графика</strong></p><p> <em>Вертикальные асимптоты:</em><br> Вертикальных асимптот нет, так как функция непрерывна на всей числовой оси.<br> <em>Горизонтальные асимптоты:</em><br> Найдем пределы функции при $x \to \pm\infty$:<br> $\lim_{x \to +\infty} (\frac{x^3}{3} + x^2 - x) = +\infty$<br> $\lim_{x \to -\infty} (\frac{x^3}{3} + x^2 - x) = -\infty$<br> Так как пределы бесконечны, горизонтальных асимптот нет.<br> <em>Наклонные асимптоты:</em><br> Наклонные асимптоты ищутся в виде $y = kx+b$.<br> $k = \lim_{x \to \infty} \frac{y(x)}{x} = \lim_{x \to \infty} \frac{\frac{x^3}{3} + x^2 - x}{x} = \lim_{x \to \infty} (\frac{x^2}{3} + x - 1) = \infty$.<br> Так как предел для $k$ равен бесконечности, наклонных асимптот нет. <br> Ответ: Асимптот нет.</p><p><strong>5. Интервалы монотонности и точки экстремума</strong></p><p> Найдем первую производную функции: <br> $y' = (\frac{x^3}{3} + x^2 - x)' = x^2 + 2x - 1$.<br> Найдем критические точки, приравняв производную к нулю: $y' = 0$.<br> $x^2 + 2x - 1 = 0$<br> $D = 2^2 - 4 \cdot 1 \cdot (-1) = 8$<br> $x_{1,2} = \frac{-2 \pm \sqrt{8}}{2} = \frac{-2 \pm 2\sqrt{2}}{2} = -1 \pm \sqrt{2}$.<br> Критические точки: $x_1 = -1 - \sqrt{2} \approx -2.41$ и $x_2 = -1 + \sqrt{2} \approx 0.41$.<br> Определим знаки производной на интервалах, образованных критическими точками:<br> - На интервале $(-\infty, -1-\sqrt{2})$: $y' > 0$, функция возрастает.<br> - На интервале $(-1-\sqrt{2}, -1+\sqrt{2})$: $y' < 0$, функция убывает.<br> - На интервале $(-1+\sqrt{2}, +\infty)$: $y' > 0$, функция возрастает.<br> В точке $x = -1 - \sqrt{2}$ производная меняет знак с `+` на `-`, следовательно, это точка локального максимума.<br> В точке $x = -1 + \sqrt{2}$ производная меняет знак с `-` на `+`, следовательно, это точка локального минимума.<br> Найдем значения функции в точках экстремума:<br> $y_{max} = y(-1-\sqrt{2}) = \frac{(-1-\sqrt{2})^3}{3} + (-1-\sqrt{2})^2 - (-1-\sqrt{2}) = \frac{5+4\sqrt{2}}{3} \approx 3.55$.<br> $y_{min} = y(-1+\sqrt{2}) = \frac{(-1+\sqrt{2})^3}{3} + (-1+\sqrt{2})^2 - (-1+\sqrt{2}) = \frac{5-4\sqrt{2}}{3} \approx -0.22$.<br> <br> Ответ: Функция возрастает на промежутках $(-\infty, -1-\sqrt{2}]$ и $[-1+\sqrt{2}, +\infty)$, убывает на промежутке $[-1-\sqrt{2}, -1+\sqrt{2}]$. Точка максимума: $(-1-\sqrt{2}, \frac{5+4\sqrt{2}}{3})$. Точка минимума: $(-1+\sqrt{2}, \frac{5-4\sqrt{2}}{3})$.</p><p><strong>6. Интервалы выпуклости/вогнутости и точки перегиба</strong></p><p> Найдем вторую производную функции:<br> $y'' = (x^2 + 2x - 1)' = 2x + 2$.<br> Найдем точки, в которых вторая производная равна нулю: $y''=0$.<br> $2x + 2 = 0 \implies x = -1$.<br> Определим знаки второй производной:<br> - При $x < -1$: $y'' < 0$, график функции выпуклый вверх (вогнутый).<br> - При $x > -1$: $y'' > 0$, график функции выпуклый вниз (выпуклый).<br> Так как в точке $x = -1$ вторая производная меняет знак, это точка перегиба.<br> Найдем значение функции в точке перегиба:<br> $y(-1) = \frac{(-1)^3}{3} + (-1)^2 - (-1) = -\frac{1}{3} + 1 + 1 = \frac{5}{3} \approx 1.67$.<br> <br> Ответ: График функции выпуклый вверх на промежутке $(-\infty, -1]$ и выпуклый вниз на промежутке $[-1, +\infty)$. Точка перегиба имеет координаты $(-1, \frac{5}{3})$.</p><p><strong>7. Построение графика</strong></p><p> Сведем полученные данные в таблицу:</p><table border="1" cellpadding="5" style="border-collapse: collapse; width: 100%;"> <tr> <td style="text-align: center;">$x$</td> <td style="text-align: center;">$(-\infty, \frac{-3-\sqrt{21}}{2})$</td> <td style="text-align: center;">$\frac{-3-\sqrt{21}}{2} \approx -3.79$</td> <td style="text-align: center;">$(\frac{-3-\sqrt{21}}{2}, -1-\sqrt{2})$</td> <td style="text-align: center;">$-1-\sqrt{2} \approx -2.41$</td> <td style="text-align: center;">$(-1-\sqrt{2}, -1)$</td> <td style="text-align: center;">$-1$</td> <td style="text-align: center;">$(-1, 0)$</td> <td style="text-align: center;">$0$</td> <td style="text-align: center;">$(0, -1+\sqrt{2})$</td> <td style="text-align: center;">$-1+\sqrt{2} \approx 0.41$</td> <td style="text-align: center;">$(-1+\sqrt{2}, \frac{-3+\sqrt{21}}{2})$</td> <td style="text-align: center;">$\frac{-3+\sqrt{21}}{2} \approx 0.79$</td> <td style="text-align: center;">$(\frac{-3+\sqrt{21}}{2}, +\infty)$</td> </tr> <tr> <td style="text-align: center;">$y'$</td> <td style="text-align: center;">+</td> <td style="text-align: center;">+</td> <td style="text-align: center;">+</td> <td style="text-align: center;">0</td> <td style="text-align: center;">-</td> <td style="text-align: center;">-</td> <td style="text-align: center;">-</td> <td style="text-align: center;">-</td> <td style="text-align: center;">-</td> <td style="text-align: center;">0</td> <td style="text-align: center;">+</td> <td style="text-align: center;">+</td> <td style="text-align: center;">+</td> </tr> <tr> <td style="text-align: center;">$y''$</td> <td style="text-align: center;">-</td> <td style="text-align: center;">-</td> <td style="text-align: center;">-</td> <td style="text-align: center;">-</td> <td style="text-align: center;">-</td> <td style="text-align: center;">0</td> <td style="text-align: center;">+</td> <td style="text-align: center;">+</td> <td style="text-align: center;">+</td> <td style="text-align: center;">+</td> <td style="text-align: center;">+</td> <td style="text-align: center;">+</td> <td style="text-align: center;">+</td> </tr> <tr> <td style="text-align: center;">$y$</td> <td style="text-align: center;">возрастает, выпукла вверх</td> <td style="text-align: center;">0 (пересечение с Ox)</td> <td style="text-align: center;">возрастает, выпукла вверх</td> <td style="text-align: center;">$\frac{5+4\sqrt{2}}{3}$ (max)</td> <td style="text-align: center;">убывает, выпукла вверх</td> <td style="text-align: center;">$\frac{5}{3}$ (перегиб)</td> <td style="text-align: center;">убывает, выпукла вниз</td> <td style="text-align: center;">0 (пересечение с осями)</td> <td style="text-align: center;">убывает, выпукла вниз</td> <td style="text-align: center;">$\frac{5-4\sqrt{2}}{3}$ (min)</td> <td style="text-align: center;">возрастает, выпукла вниз</td> <td style="text-align: center;">0 (пересечение с Ox)</td> <td style="text-align: center;">возрастает, выпукла вниз</td> </tr></table><p> Основываясь на проведенном исследовании, можно построить график функции. График представляет собой кубическую параболу, которая начинается в $-\infty$, возрастает до точки максимума $(-2.41, 3.55)$, затем убывает, проходя через точку перегиба $(-1, 1.67)$ и точку начала координат $(0,0)$, достигает минимума в точке $(0.41, -0.22)$, после чего снова возрастает и уходит в $+\infty$. График пересекает ось абсцисс в трех точках: $x \approx -3.79$, $x=0$ и $x \approx 0.79$. <br> Ответ: Полное исследование функции проведено, ключевые точки и интервалы поведения функции найдены, что позволяет построить ее график.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1120285" "type" => "task" ] "previous" => array:2 [ "refs" => "1120283" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1135 #items: array:1 [ 0 => App\Models\Book {#1158} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1141 #items: array:1 [ 0 => App\Models\BookPage {#1108 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" 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#escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1106 #items: array:1 [ 0 => App\Models\Element {#1105 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1099226" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1099224" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1360 #items: array:13 [ 0 => App\Models\Task {#1380 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Task {#1385 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Task {#1389 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 3 => App\Models\Task {#1397 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 4 => App\Models\Task {#1418 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 5 => App\Models\Task {#1430 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 6 => App\Models\Task {#1442 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 7 => App\Models\Task {#1454 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 8 => App\Models\Task {#1466 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 9 => App\Models\Task {#1478 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 10 => App\Models\Task {#1490 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] 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#fillable: [] #guarded: array:1 [ …1] } 12 => App\Models\Task {#1514 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1099225 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "72" "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/page-2-72" "field_display_title" => "72" "field_folder" => "2" "field_image_name" => "72" "field_branch_parent" => 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13:58:26" "updated_at" => null "field_page_start" => "72" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/5-3-2zad-26" "field_display_title" => "26" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1072} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1143} "content" => Illuminate\Database\Eloquent\Collection {#1104} "next" => array:2 [ "refs" => "1120285" "type" => "task" ] "previous" => array:2 [ "refs" => "1120283" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1135} "page" => array:2 [ "refs" => "1099225" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№26 (с. 72)
Условие. №26 (с. 72)
скриншот условия
26.
(2) $y = \frac{x^3}{3} + x^2 - x$
Решение 2 (rus). №26 (с. 72)
Проведем полное исследование функции $y = \frac{x^3}{3} + x^2 - x$ и построим ее график.
1. Область определения
Функция является многочленом, поэтому она определена для всех действительных чисел.
Ответ: Область определения функции $D(y) = (-\infty, +\infty)$.
2. Точки пересечения с осями координат
С осью Oy:
Для нахождения точки пересечения с осью ординат, подставим $x=0$ в уравнение функции:
$y(0) = \frac{0^3}{3} + 0^2 - 0 = 0$.
Точка пересечения с осью Oy: $(0, 0)$.
С осью Ox:
Для нахождения точек пересечения с осью абсцисс, решим уравнение $y=0$:
$\frac{x^3}{3} + x^2 - x = 0$
Вынесем $x$ за скобки:
$x(\frac{x^2}{3} + x - 1) = 0$
Отсюда получаем первый корень $x_1 = 0$.
Решим квадратное уравнение $\frac{x^2}{3} + x - 1 = 0$. Умножим обе части на 3:
$x^2 + 3x - 3 = 0$
Найдем дискриминант: $D = b^2 - 4ac = 3^2 - 4 \cdot 1 \cdot (-3) = 9 + 12 = 21$.
Корни уравнения: $x_{2,3} = \frac{-3 \pm \sqrt{21}}{2}$.
$x_2 = \frac{-3 - \sqrt{21}}{2} \approx -3.79$, $x_3 = \frac{-3 + \sqrt{21}}{2} \approx 0.79$.
Ответ: Точка пересечения с осью Oy: $(0, 0)$. Точки пересечения с осью Ox: $(0, 0)$, $(\frac{-3 - \sqrt{21}}{2}, 0)$, $(\frac{-3 + \sqrt{21}}{2}, 0)$.
3. Четность и периодичность
Проверим функцию на четность: $y(-x) = \frac{(-x)^3}{3} + (-x)^2 - (-x) = -\frac{x^3}{3} + x^2 + x$.
Так как $y(-x) \neq y(x)$ и $y(-x) \neq -y(x)$, функция не является ни четной, ни нечетной. Это функция общего вида.
Функция не является периодической, так как является многочленом.
Ответ: Функция общего вида, непериодическая.
4. Асимптоты графика
Вертикальные асимптоты:
Вертикальных асимптот нет, так как функция непрерывна на всей числовой оси.
Горизонтальные асимптоты:
Найдем пределы функции при $x \to \pm\infty$:
$\lim_{x \to +\infty} (\frac{x^3}{3} + x^2 - x) = +\infty$
$\lim_{x \to -\infty} (\frac{x^3}{3} + x^2 - x) = -\infty$
Так как пределы бесконечны, горизонтальных асимптот нет.
Наклонные асимптоты:
Наклонные асимптоты ищутся в виде $y = kx+b$.
$k = \lim_{x \to \infty} \frac{y(x)}{x} = \lim_{x \to \infty} \frac{\frac{x^3}{3} + x^2 - x}{x} = \lim_{x \to \infty} (\frac{x^2}{3} + x - 1) = \infty$.
Так как предел для $k$ равен бесконечности, наклонных асимптот нет.
Ответ: Асимптот нет.
5. Интервалы монотонности и точки экстремума
Найдем первую производную функции:
$y' = (\frac{x^3}{3} + x^2 - x)' = x^2 + 2x - 1$.
Найдем критические точки, приравняв производную к нулю: $y' = 0$.
$x^2 + 2x - 1 = 0$
$D = 2^2 - 4 \cdot 1 \cdot (-1) = 8$
$x_{1,2} = \frac{-2 \pm \sqrt{8}}{2} = \frac{-2 \pm 2\sqrt{2}}{2} = -1 \pm \sqrt{2}$.
Критические точки: $x_1 = -1 - \sqrt{2} \approx -2.41$ и $x_2 = -1 + \sqrt{2} \approx 0.41$.
Определим знаки производной на интервалах, образованных критическими точками:
- На интервале $(-\infty, -1-\sqrt{2})$: $y' > 0$, функция возрастает.
- На интервале $(-1-\sqrt{2}, -1+\sqrt{2})$: $y' < 0$, функция убывает.
- На интервале $(-1+\sqrt{2}, +\infty)$: $y' > 0$, функция возрастает.
В точке $x = -1 - \sqrt{2}$ производная меняет знак с `+` на `-`, следовательно, это точка локального максимума.
В точке $x = -1 + \sqrt{2}$ производная меняет знак с `-` на `+`, следовательно, это точка локального минимума.
Найдем значения функции в точках экстремума:
$y_{max} = y(-1-\sqrt{2}) = \frac{(-1-\sqrt{2})^3}{3} + (-1-\sqrt{2})^2 - (-1-\sqrt{2}) = \frac{5+4\sqrt{2}}{3} \approx 3.55$.
$y_{min} = y(-1+\sqrt{2}) = \frac{(-1+\sqrt{2})^3}{3} + (-1+\sqrt{2})^2 - (-1+\sqrt{2}) = \frac{5-4\sqrt{2}}{3} \approx -0.22$.
Ответ: Функция возрастает на промежутках $(-\infty, -1-\sqrt{2}]$ и $[-1+\sqrt{2}, +\infty)$, убывает на промежутке $[-1-\sqrt{2}, -1+\sqrt{2}]$. Точка максимума: $(-1-\sqrt{2}, \frac{5+4\sqrt{2}}{3})$. Точка минимума: $(-1+\sqrt{2}, \frac{5-4\sqrt{2}}{3})$.
6. Интервалы выпуклости/вогнутости и точки перегиба
Найдем вторую производную функции:
$y'' = (x^2 + 2x - 1)' = 2x + 2$.
Найдем точки, в которых вторая производная равна нулю: $y''=0$.
$2x + 2 = 0 \implies x = -1$.
Определим знаки второй производной:
- При $x < -1$: $y'' < 0$, график функции выпуклый вверх (вогнутый).
- При $x > -1$: $y'' > 0$, график функции выпуклый вниз (выпуклый).
Так как в точке $x = -1$ вторая производная меняет знак, это точка перегиба.
Найдем значение функции в точке перегиба:
$y(-1) = \frac{(-1)^3}{3} + (-1)^2 - (-1) = -\frac{1}{3} + 1 + 1 = \frac{5}{3} \approx 1.67$.
Ответ: График функции выпуклый вверх на промежутке $(-\infty, -1]$ и выпуклый вниз на промежутке $[-1, +\infty)$. Точка перегиба имеет координаты $(-1, \frac{5}{3})$.
7. Построение графика
Сведем полученные данные в таблицу:
| $x$ | $(-\infty, \frac{-3-\sqrt{21}}{2})$ | $\frac{-3-\sqrt{21}}{2} \approx -3.79$ | $(\frac{-3-\sqrt{21}}{2}, -1-\sqrt{2})$ | $-1-\sqrt{2} \approx -2.41$ | $(-1-\sqrt{2}, -1)$ | $-1$ | $(-1, 0)$ | $0$ | $(0, -1+\sqrt{2})$ | $-1+\sqrt{2} \approx 0.41$ | $(-1+\sqrt{2}, \frac{-3+\sqrt{21}}{2})$ | $\frac{-3+\sqrt{21}}{2} \approx 0.79$ | $(\frac{-3+\sqrt{21}}{2}, +\infty)$ |
| $y'$ | + | + | + | 0 | - | - | - | - | - | 0 | + | + | + |
| $y''$ | - | - | - | - | - | 0 | + | + | + | + | + | + | + |
| $y$ | возрастает, выпукла вверх | 0 (пересечение с Ox) | возрастает, выпукла вверх | $\frac{5+4\sqrt{2}}{3}$ (max) | убывает, выпукла вверх | $\frac{5}{3}$ (перегиб) | убывает, выпукла вниз | 0 (пересечение с осями) | убывает, выпукла вниз | $\frac{5-4\sqrt{2}}{3}$ (min) | возрастает, выпукла вниз | 0 (пересечение с Ox) | возрастает, выпукла вниз |
Основываясь на проведенном исследовании, можно построить график функции. График представляет собой кубическую параболу, которая начинается в $-\infty$, возрастает до точки максимума $(-2.41, 3.55)$, затем убывает, проходя через точку перегиба $(-1, 1.67)$ и точку начала координат $(0,0)$, достигает минимума в точке $(0.41, -0.22)$, после чего снова возрастает и уходит в $+\infty$. График пересекает ось абсцисс в трех точках: $x \approx -3.79$, $x=0$ и $x \approx 0.79$.
Ответ: Полное исследование функции проведено, ключевые точки и интервалы поведения функции найдены, что позволяет построить ее график.
Другие задания:
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