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Номер 14, страница 91, 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 6. Применение производной. Параграф 2. Экстремумы и критические точки - номер 14, страница 91.
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" => 1120351 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "91" "field_page_end" => "92" "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-2zad-14" "field_display_title" => "14" "field_outside_task" => "0" "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:3 [ 0 => App\Models\Branch {#1153 #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 {#1157 #items: array:1 [ 0 => App\Models\Term {#1154 #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 {#1186 #items: array:1 [ 0 => App\Models\Book {#1156 #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 {#1158 #items: array:1 [ 0 => App\Models\Term {#1155 #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 {#1159 #items: array:1 [ 0 => App\Models\Term {#1160 #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 {#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" => 7006 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматыкітап баспасы" "field_cases" => array:6 [ …6] "field_translit" => "almatykitap baspasy" ] #original: array:6 [ "id" => 7006 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматыкітап баспасы" "field_cases" => array:6 [ …6] "field_translit" => "almatykitap baspasy" ] #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_author" => Illuminate\Database\Eloquent\Collection {#1163 #items: array:3 [ 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:12 [ "id" => 7007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Пак" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Олег" "field_patronymic" => "Владимирович" "field_surname_rp" => null ] #original: array:12 [ "id" => 7007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Пак" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Олег" "field_patronymic" => "Владимирович" "field_surname_rp" => null ] #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\Term {#1165 #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 [ "id" => 7008 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ардакулы" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Дархан" "field_patronymic" => null "field_surname_rp" => null ] #original: array:12 [ "id" => 7008 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ардакулы" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Дархан" "field_patronymic" => null "field_surname_rp" => null ] #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\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 [ "id" => 7009 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ескендирова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Елена" "field_patronymic" => "Викторовна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7009 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ескендирова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Елена" "field_patronymic" => "Викторовна" "field_surname_rp" => 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_author_foreign" => Illuminate\Database\Eloquent\Collection {#1167 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1168 #items: array:1 [ 0 => App\Models\Term {#1169 #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" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #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_country" => 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:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #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_city" => 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 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #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_series" => Illuminate\Database\Eloquent\Collection {#1174 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1175 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1176 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1177 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1178 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1179 #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 {#1180 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1181 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1182 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1183 #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 {#1158} "field_class" => Illuminate\Database\Eloquent\Collection {#1159} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1161} "field_author" => Illuminate\Database\Eloquent\Collection {#1163} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1167} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1168} "field_country" => Illuminate\Database\Eloquent\Collection {#1170} "field_city" => Illuminate\Database\Eloquent\Collection {#1172} "field_series" => Illuminate\Database\Eloquent\Collection {#1174} "field_umk" => Illuminate\Database\Eloquent\Collection {#1175} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1176} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1177} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1178} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1179} "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 {#1180} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1181} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1182} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1183} "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 {#1184 #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 {#1157} "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 {#1186} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1184} ] #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 {#1185 #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" => 1119442 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Применение производной" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1190 #items: array:1 [ 0 => App\Models\Term {#1187 #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" => "76" "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 {#1189 #items: array:1 [ 0 => App\Models\Book {#1156} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1188 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119442 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Применение производной" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1190} "field_page_start" => "76" "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 {#1189} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1188} ] #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 {#1191 #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" => 1119446 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Экстремумы и критические точки" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1195 #items: array:1 [ 0 => App\Models\Term {#1192 #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" => "86" "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 {#1194 #items: array:1 [ 0 => App\Models\Book {#1156} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1235 #items: array:1 [ 0 => 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" => 1119442 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Применение производной" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1197 …2} "field_page_start" => "76" "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 {#1198 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1227 …2} ] #original: array:24 [ "id" => 1119442 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Применение производной" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1197 …2} "field_page_start" => "76" "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 {#1198 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1227 …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" => 1119446 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Экстремумы и критические точки" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1195} "field_page_start" => "86" "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 {#1194} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1235} ] #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" => 1301036 "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" => "1120351" "type" => "task" ] "text" => "<p><strong>14. (1)</strong> На рисунке 5 изображен график функции $y=f(x)$, $D(f):[-7;7]$.</p><svg width="400" height="400" viewBox="-7.5 -7.5 15 15"><!-- Grid lines --><g stroke="#ccc" stroke-width="0.05"><!-- Vertical lines --><line x1="-7" y1="-7" x2="-7" y2="7" /><line x1="-6" y1="-7" x2="-6" y2="7" /><line x1="-5" y1="-7" x2="-5" y2="7" /><line x1="-4" y1="-7" x2="-4" y2="7" /><line x1="-3" y1="-7" x2="-3" y2="7" /><line x1="-2" y1="-7" x2="-2" y2="7" /><line x1="-1" y1="-7" x2="-1" y2="7" /><line x1="1" y1="-7" x2="1" y2="7" /><line x1="2" y1="-7" x2="2" y2="7" /><line x1="3" y1="-7" x2="3" y2="7" /><line x1="4" y1="-7" x2="4" y2="7" /><line x1="5" y1="-7" x2="5" y2="7" /><line x1="6" y1="-7" x2="6" y2="7" /><line x1="7" y1="-7" x2="7" y2="7" /><!-- Horizontal lines --><line x1="-7" y1="-7" x2="7" y2="-7" /><line x1="-7" y1="-6" x2="7" y2="-6" /><line x1="-7" y1="-5" x2="7" y2="-5" /><line x1="-7" y1="-4" x2="7" y2="-4" /><line x1="-7" y1="-3" x2="7" y2="-3" /><line x1="-7" y1="-2" x2="7" y2="-2" /><line x1="-7" y1="-1" x2="7" y2="-1" /><line x1="-7" y1="1" x2="7" y2="1" /><line x1="-7" y1="2" x2="7" y2="2" /><line x1="-7" y1="3" x2="7" y2="3" /><line x1="-7" y1="4" x2="7" y2="4" /><line x1="-7" y1="5" x2="7" y2="5" /><line x1="-7" y1="6" x2="7" y2="6" /><line x1="-7" y1="7" x2="7" y2="7" /></g><!-- Axes --><line x1="-7.2" y1="0" x2="7.2" y2="0" stroke="black" stroke-width="0.1" marker-end="url(#arrowhead)"/><line x1="0" y1="-7.2" x2="0" y2="7.2" stroke="black" stroke-width="0.1" marker-end="url(#arrowhead)"/><!-- Arrowheads for axes --><defs><marker id="arrowhead" markerWidth="0.8" markerHeight="0.8" refX="0.4" refY="0.4" orient="auto" markerUnits="strokeWidth"><path d="M0,0 L0,0.8 L0.8,0.4 Z" fill="black" /></marker></defs><!-- Origin label --><text x="0.2" y="-0.3" font-size="0.6" text-anchor="start">O</text><!-- X and Y labels --><text x="7.3" y="-0.3" font-size="0.6" text-anchor="end">x</text><text x="0.3" y="7.3" font-size="0.6" text-anchor="start">y</text><!-- Dashed line for tangent --><line x1="-1.5" y1="2" x2="2.5" y2="2" stroke="gray" stroke-width="0.08" stroke-dasharray="0.2,0.2" /><text x="-1.5" y="2.7" font-size="0.4" fill="black">касательная в т. x=2</text><!-- Function curve --><path d="M -7 -1 C -6.5 0.5 -5.5 1 -5 1 S -4.5 0 -4 0 S -3.5 -1 -3 -1 S -2.5 0 -2 0 S -1.5 1 -1 2 S -0.5 3 0 3 S 0.5 2.5 1 2.5 S 1.5 2 2 2 S 2.5 1.5 3 1 S 3.5 1.5 4 1.5 S 4.5 1.25 5 1 S 5.5 1.75 6 2 S 6.5 1.5 7 1" fill="none" stroke="blue" stroke-width="0.15" /></svg><p>Рис. 5</p><p><strong>а) Укажите все критические точки функции $f(x)$.</strong></p><p><strong>б) Укажите точки локальных экстремумов функции $f(x)$.</strong></p><p><strong>в) Решите неравенство $f'(x) > 0$, $f'(x) \ge 0$, $f'(x) < 0$, $f'(x) \le 0$.</strong></p>" "img" => array:2 [ 0 => array:5 [ "name" => "14-1.jpg" "alt" => null "width" => "1595" "height" => 734 "path" => "/media/algebra_10/baspasy-u19/0-6-2zad/14-1.webp?ts=1753534906" ] 1 => array:5 [ "name" => "14-2.jpg" "alt" => null "width" => "1425" "height" => 290 "path" => "/media/algebra_10/baspasy-u19/0-6-2zad/14-2.webp?ts=1753534906" ] ] ] #original: array:7 [ "id" => 1301036 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1122} "task" => array:2 [ "refs" => "1120351" "type" => "task" ] "text" => "<p><strong>14. (1)</strong> На рисунке 5 изображен график функции $y=f(x)$, $D(f):[-7;7]$.</p><svg width="400" height="400" viewBox="-7.5 -7.5 15 15"><!-- Grid lines --><g stroke="#ccc" stroke-width="0.05"><!-- Vertical lines --><line x1="-7" y1="-7" x2="-7" y2="7" /><line x1="-6" y1="-7" x2="-6" y2="7" /><line x1="-5" y1="-7" x2="-5" y2="7" /><line x1="-4" y1="-7" x2="-4" y2="7" /><line x1="-3" y1="-7" x2="-3" y2="7" /><line x1="-2" y1="-7" x2="-2" y2="7" /><line x1="-1" y1="-7" x2="-1" y2="7" /><line x1="1" y1="-7" x2="1" y2="7" /><line x1="2" y1="-7" x2="2" y2="7" /><line x1="3" y1="-7" x2="3" y2="7" /><line x1="4" y1="-7" x2="4" y2="7" /><line x1="5" y1="-7" x2="5" y2="7" /><line x1="6" y1="-7" x2="6" y2="7" /><line x1="7" y1="-7" x2="7" y2="7" /><!-- Horizontal lines --><line x1="-7" y1="-7" x2="7" y2="-7" /><line x1="-7" y1="-6" x2="7" y2="-6" /><line x1="-7" y1="-5" x2="7" y2="-5" /><line x1="-7" y1="-4" x2="7" y2="-4" /><line x1="-7" y1="-3" x2="7" y2="-3" /><line x1="-7" y1="-2" x2="7" y2="-2" /><line x1="-7" y1="-1" x2="7" y2="-1" /><line x1="-7" y1="1" x2="7" y2="1" /><line x1="-7" y1="2" x2="7" y2="2" /><line x1="-7" y1="3" x2="7" y2="3" /><line x1="-7" y1="4" x2="7" y2="4" /><line x1="-7" y1="5" x2="7" y2="5" /><line x1="-7" y1="6" x2="7" y2="6" /><line x1="-7" y1="7" x2="7" y2="7" /></g><!-- Axes --><line x1="-7.2" y1="0" x2="7.2" y2="0" stroke="black" stroke-width="0.1" marker-end="url(#arrowhead)"/><line x1="0" y1="-7.2" x2="0" y2="7.2" stroke="black" stroke-width="0.1" marker-end="url(#arrowhead)"/><!-- Arrowheads for axes --><defs><marker id="arrowhead" markerWidth="0.8" markerHeight="0.8" refX="0.4" refY="0.4" orient="auto" markerUnits="strokeWidth"><path d="M0,0 L0,0.8 L0.8,0.4 Z" fill="black" /></marker></defs><!-- Origin label --><text x="0.2" y="-0.3" font-size="0.6" text-anchor="start">O</text><!-- X and Y labels --><text x="7.3" y="-0.3" font-size="0.6" text-anchor="end">x</text><text x="0.3" y="7.3" font-size="0.6" text-anchor="start">y</text><!-- Dashed line for tangent --><line x1="-1.5" y1="2" x2="2.5" y2="2" stroke="gray" stroke-width="0.08" stroke-dasharray="0.2,0.2" /><text x="-1.5" y="2.7" font-size="0.4" fill="black">касательная в т. x=2</text><!-- Function curve --><path d="M -7 -1 C -6.5 0.5 -5.5 1 -5 1 S -4.5 0 -4 0 S -3.5 -1 -3 -1 S -2.5 0 -2 0 S -1.5 1 -1 2 S -0.5 3 0 3 S 0.5 2.5 1 2.5 S 1.5 2 2 2 S 2.5 1.5 3 1 S 3.5 1.5 4 1.5 S 4.5 1.25 5 1 S 5.5 1.75 6 2 S 6.5 1.5 7 1" fill="none" stroke="blue" stroke-width="0.15" /></svg><p>Рис. 5</p><p><strong>а) Укажите все критические точки функции $f(x)$.</strong></p><p><strong>б) Укажите точки локальных экстремумов функции $f(x)$.</strong></p><p><strong>в) Решите неравенство $f'(x) > 0$, $f'(x) \ge 0$, $f'(x) < 0$, $f'(x) \le 0$.</strong></p>" "img" => array:2 [ 0 => array:5 [ "name" => "14-1.jpg" "alt" => null "width" => "1595" "height" => 734 "path" => "/media/algebra_10/baspasy-u19/0-6-2zad/14-1.webp?ts=1753534906" ] 1 => array:5 [ "name" => "14-2.jpg" "alt" => null "width" => "1425" "height" => 290 "path" => "/media/algebra_10/baspasy-u19/0-6-2zad/14-2.webp?ts=1753534906" ] ] ] #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" => 1552072 "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" => "1120351" "type" => "task" ] "text" => "<p><strong>а) Укажите все критические точки функции f(x).</strong></p><p>Критическими точками функции называются внутренние точки области определения, в которых производная функции равна нулю или не существует. Область определения функции, указанная в условии, это $D(f) = [-7; 7]$, следовательно, мы ищем критические точки на интервале $(-7; 7)$.</p><p>По графику видно, что функция $f(x)$ является гладкой (дифференцируемой) на всем интервале $(-7; 7)$, поэтому точек, где производная $f'(x)$ не существует, нет. Критическими точками будут только те точки, в которых производная равна нулю ($f'(x) = 0$). Геометрически это означает, что касательная к графику функции в этих точках горизонтальна.</p><p>На графике мы можем определить следующие точки, где касательная горизонтальна:<br>– Точки локальных минимумов: $x = -5, x = -1, x = 4$.<br>– Точки локальных максимумов: $x = -3, x = 1, x = 6$.<br>– Точка перегиба, в которой касательная горизонтальна (это особо отмечено на рисунке пунктирной линией): $x = 2$.</p><p>Таким образом, все эти точки являются критическими.</p><p><strong>Ответ:</strong> -5; -3; -1; 1; 2; 4; 6.</p><p><strong>б) Укажите точки локальных экстремумов функции f(x).</strong></p><p>Точки локальных экстремумов – это точки, в которых функция достигает своего локального (местного) максимума или минимума. В этих точках происходит смена характера монотонности функции (с возрастания на убывание или наоборот).</p><p>Анализируя график, находим:<br>– Точки локального максимума (соответствуют "вершинам" на графике, где возрастание сменяется убыванием): $x = -3, x = 1, x = 6$.<br>– Точки локального минимума (соответствуют "впадинам" на графике, где убывание сменяется возрастанием): $x = -5, x = -1, x = 4$.</p><p>Точка $x=2$ не является точкой экстремума. Несмотря на то, что производная в этой точке равна нулю, функция убывает как до, так и после этой точки. Это точка перегиба.</p><p><strong>Ответ:</strong> Точки локального максимума: -3; 1; 6. Точки локального минимума: -5; -1; 4.</p><p><strong>в) Решите неравенство f'(x) > 0, f'(x) ≥ 0, f'(x) < 0, f'(x) ≤ 0.</strong></p><p>Знак производной функции $f'(x)$ напрямую связан с поведением самой функции $f(x)$:<br>– если $f'(x) > 0$, функция $f(x)$ строго возрастает;<br>– если $f'(x) < 0$, функция $f(x)$ строго убывает.<br>Решим каждое неравенство, анализируя график функции на области определения $[-7; 7]$.</p><p><b>Решение для $f'(x) > 0$:</b><br>Неравенство выполняется на интервалах, где функция $f(x)$ строго возрастает.<br>По графику это интервалы $(-5, -3)$, $(-1, 1)$ и $(4, 6)$.</p><p><b>Решение для $f'(x) \ge 0$:</b><br>Неравенство выполняется на промежутках, где функция $f(x)$ не убывает (т.е. возрастает или имеет горизонтальную касательную). Для этого к интервалам строгого возрастания нужно добавить точки, где $f'(x)=0$.<br>Получаем отрезки $[-5, -3]$, $[-1, 1]$ и $[4, 6]$. Также необходимо учесть изолированную точку $x=2$, где $f'(2)=0$, которая тоже удовлетворяет неравенству.</p><p><b>Решение для $f'(x) < 0$:</b><br>Неравенство выполняется на интервалах, где функция $f(x)$ строго убывает.<br>По графику это интервалы $(-7, -5)$, $(-3, -1)$, а также $(1, 4)$ и $(6, 7)$. Однако в точке $x=2$ производная равна нулю, поэтому эту точку нужно исключить. Таким образом, получаем объединение интервалов: $(-7, -5) \cup (-3, -1) \cup (1, 2) \cup (2, 4) \cup (6, 7)$.</p><p><b>Решение для $f'(x) \le 0$:</b><br>Неравенство выполняется на промежутках, где функция $f(x)$ не возрастает (т.е. убывает или имеет горизонтальную касательную). Для этого к интервалам строгого убывания нужно добавить точки, где $f'(x)=0$.<br>Получаем отрезки $[-7, -5]$, $[-3, -1]$, $[1, 4]$ (этот отрезок включает точку $x=2$, где $f'(2)=0$) и $[6, 7]$.</p><p>Ответ:<br>$f'(x) > 0$ при $x \in (-5, -3) \cup (-1, 1) \cup (4, 6)$.<br>$f'(x) \ge 0$ при $x \in [-5, -3] \cup [-1, 1] \cup [4, 6] \cup \{2\}$.<br>$f'(x) < 0$ при $x \in (-7, -5) \cup (-3, -1) \cup (1, 2) \cup (2, 4) \cup (6, 7)$.<br>$f'(x) \le 0$ при $x \in [-7, -5] \cup [-3, -1] \cup [1, 4] \cup [6, 7]$.</p>" ] #original: array:6 [ "id" => 1552072 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116} "task" => array:2 [ "refs" => "1120351" "type" => "task" ] "text" => "<p><strong>а) Укажите все критические точки функции f(x).</strong></p><p>Критическими точками функции называются внутренние точки области определения, в которых производная функции равна нулю или не существует. Область определения функции, указанная в условии, это $D(f) = [-7; 7]$, следовательно, мы ищем критические точки на интервале $(-7; 7)$.</p><p>По графику видно, что функция $f(x)$ является гладкой (дифференцируемой) на всем интервале $(-7; 7)$, поэтому точек, где производная $f'(x)$ не существует, нет. Критическими точками будут только те точки, в которых производная равна нулю ($f'(x) = 0$). Геометрически это означает, что касательная к графику функции в этих точках горизонтальна.</p><p>На графике мы можем определить следующие точки, где касательная горизонтальна:<br>– Точки локальных минимумов: $x = -5, x = -1, x = 4$.<br>– Точки локальных максимумов: $x = -3, x = 1, x = 6$.<br>– Точка перегиба, в которой касательная горизонтальна (это особо отмечено на рисунке пунктирной линией): $x = 2$.</p><p>Таким образом, все эти точки являются критическими.</p><p><strong>Ответ:</strong> -5; -3; -1; 1; 2; 4; 6.</p><p><strong>б) Укажите точки локальных экстремумов функции f(x).</strong></p><p>Точки локальных экстремумов – это точки, в которых функция достигает своего локального (местного) максимума или минимума. В этих точках происходит смена характера монотонности функции (с возрастания на убывание или наоборот).</p><p>Анализируя график, находим:<br>– Точки локального максимума (соответствуют "вершинам" на графике, где возрастание сменяется убыванием): $x = -3, x = 1, x = 6$.<br>– Точки локального минимума (соответствуют "впадинам" на графике, где убывание сменяется возрастанием): $x = -5, x = -1, x = 4$.</p><p>Точка $x=2$ не является точкой экстремума. Несмотря на то, что производная в этой точке равна нулю, функция убывает как до, так и после этой точки. Это точка перегиба.</p><p><strong>Ответ:</strong> Точки локального максимума: -3; 1; 6. Точки локального минимума: -5; -1; 4.</p><p><strong>в) Решите неравенство f'(x) > 0, f'(x) ≥ 0, f'(x) < 0, f'(x) ≤ 0.</strong></p><p>Знак производной функции $f'(x)$ напрямую связан с поведением самой функции $f(x)$:<br>– если $f'(x) > 0$, функция $f(x)$ строго возрастает;<br>– если $f'(x) < 0$, функция $f(x)$ строго убывает.<br>Решим каждое неравенство, анализируя график функции на области определения $[-7; 7]$.</p><p><b>Решение для $f'(x) > 0$:</b><br>Неравенство выполняется на интервалах, где функция $f(x)$ строго возрастает.<br>По графику это интервалы $(-5, -3)$, $(-1, 1)$ и $(4, 6)$.</p><p><b>Решение для $f'(x) \ge 0$:</b><br>Неравенство выполняется на промежутках, где функция $f(x)$ не убывает (т.е. возрастает или имеет горизонтальную касательную). Для этого к интервалам строгого возрастания нужно добавить точки, где $f'(x)=0$.<br>Получаем отрезки $[-5, -3]$, $[-1, 1]$ и $[4, 6]$. Также необходимо учесть изолированную точку $x=2$, где $f'(2)=0$, которая тоже удовлетворяет неравенству.</p><p><b>Решение для $f'(x) < 0$:</b><br>Неравенство выполняется на интервалах, где функция $f(x)$ строго убывает.<br>По графику это интервалы $(-7, -5)$, $(-3, -1)$, а также $(1, 4)$ и $(6, 7)$. Однако в точке $x=2$ производная равна нулю, поэтому эту точку нужно исключить. Таким образом, получаем объединение интервалов: $(-7, -5) \cup (-3, -1) \cup (1, 2) \cup (2, 4) \cup (6, 7)$.</p><p><b>Решение для $f'(x) \le 0$:</b><br>Неравенство выполняется на промежутках, где функция $f(x)$ не возрастает (т.е. убывает или имеет горизонтальную касательную). Для этого к интервалам строгого убывания нужно добавить точки, где $f'(x)=0$.<br>Получаем отрезки $[-7, -5]$, $[-3, -1]$, $[1, 4]$ (этот отрезок включает точку $x=2$, где $f'(2)=0$) и $[6, 7]$.</p><p>Ответ:<br>$f'(x) > 0$ при $x \in (-5, -3) \cup (-1, 1) \cup (4, 6)$.<br>$f'(x) \ge 0$ при $x \in [-5, -3] \cup [-1, 1] \cup [4, 6] \cup \{2\}$.<br>$f'(x) < 0$ при $x \in (-7, -5) \cup (-3, -1) \cup (1, 2) \cup (2, 4) \cup (6, 7)$.<br>$f'(x) \le 0$ при $x \in [-7, -5] \cup [-3, -1] \cup [1, 4] \cup [6, 7]$.</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" => "1120352" "type" => "task" ] "previous" => array:2 [ "refs" => "1120350" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1135 #items: array:1 [ 0 => App\Models\Book {#1156} ] #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" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1099244 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "91" "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/page-2-91" "field_display_title" => "91" "field_folder" => "2" "field_image_name" => "91" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1118 #items: array:1 [ 0 => App\Models\Branch {#1153} ] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1117 #items: array:1 [ 0 => App\Models\Book {#1156} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1113 #items: [] #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 [ "id" => 1262669 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1070 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1262669 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1070 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #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" => "1099245" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1099243" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1327 #items: array:8 [ 0 => App\Models\Task {#1341 #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" => 1120344 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "90" "field_page_end" => "91" "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-2zad-7" "field_display_title" => "7" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1342 …2} "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 {#1343 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1344 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1345 …2} "content" => Illuminate\Database\Eloquent\Collection {#1347 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1353 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1120344 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "90" "field_page_end" => "91" "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-2zad-7" "field_display_title" => "7" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1342 …2} "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 {#1343 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1344 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1345 …2} "content" => Illuminate\Database\Eloquent\Collection {#1347 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1353 …2} "page" => array:2 [ …2] ] #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\Task {#1348 #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" => 1120345 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "91" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-2zad-8" "field_display_title" => "8" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1346 …2} "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 {#1355 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1351 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1349 …2} "content" => Illuminate\Database\Eloquent\Collection {#1350 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1365 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1120345 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "91" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-2zad-8" "field_display_title" => "8" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1346 …2} "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 {#1355 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1351 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1349 …2} "content" => Illuminate\Database\Eloquent\Collection {#1350 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1365 …2} "page" => array:2 [ …2] ] #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\Task {#1360 #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" => 1120346 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "91" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-2zad-9" "field_display_title" => "9" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1352 …2} "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 {#1367 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1363 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1361 …2} "content" => Illuminate\Database\Eloquent\Collection {#1362 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1377 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1120346 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "91" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-2zad-9" "field_display_title" => "9" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1352 …2} "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 {#1367 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1363 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1361 …2} "content" => Illuminate\Database\Eloquent\Collection {#1362 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1377 …2} "page" => array:2 [ …2] ] #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\Task {#1372 #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" => 1120347 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "91" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-2zad-10" "field_display_title" => "10" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1364 …2} "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 {#1379 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1375 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1373 …2} "content" => Illuminate\Database\Eloquent\Collection {#1374 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1389 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1120347 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "91" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-2zad-10" "field_display_title" => "10" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1364 …2} "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 {#1379 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1375 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1373 …2} "content" => Illuminate\Database\Eloquent\Collection {#1374 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1389 …2} "page" => array:2 [ …2] ] #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\Task {#1384 #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" => 1120348 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "91" 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№14 (с. 91)
Условие. №14 (с. 91)
скриншот условия
14. (1) На рисунке 5 изображен график функции $y=f(x)$, $D(f):[-7;7]$.
Рис. 5
а) Укажите все критические точки функции $f(x)$.
б) Укажите точки локальных экстремумов функции $f(x)$.
в) Решите неравенство $f'(x) > 0$, $f'(x) \ge 0$, $f'(x) < 0$, $f'(x) \le 0$.
Решение 2 (rus). №14 (с. 91)
а) Укажите все критические точки функции f(x).
Критическими точками функции называются внутренние точки области определения, в которых производная функции равна нулю или не существует. Область определения функции, указанная в условии, это $D(f) = [-7; 7]$, следовательно, мы ищем критические точки на интервале $(-7; 7)$.
По графику видно, что функция $f(x)$ является гладкой (дифференцируемой) на всем интервале $(-7; 7)$, поэтому точек, где производная $f'(x)$ не существует, нет. Критическими точками будут только те точки, в которых производная равна нулю ($f'(x) = 0$). Геометрически это означает, что касательная к графику функции в этих точках горизонтальна.
На графике мы можем определить следующие точки, где касательная горизонтальна:
– Точки локальных минимумов: $x = -5, x = -1, x = 4$.
– Точки локальных максимумов: $x = -3, x = 1, x = 6$.
– Точка перегиба, в которой касательная горизонтальна (это особо отмечено на рисунке пунктирной линией): $x = 2$.
Таким образом, все эти точки являются критическими.
Ответ: -5; -3; -1; 1; 2; 4; 6.
б) Укажите точки локальных экстремумов функции f(x).
Точки локальных экстремумов – это точки, в которых функция достигает своего локального (местного) максимума или минимума. В этих точках происходит смена характера монотонности функции (с возрастания на убывание или наоборот).
Анализируя график, находим:
– Точки локального максимума (соответствуют "вершинам" на графике, где возрастание сменяется убыванием): $x = -3, x = 1, x = 6$.
– Точки локального минимума (соответствуют "впадинам" на графике, где убывание сменяется возрастанием): $x = -5, x = -1, x = 4$.
Точка $x=2$ не является точкой экстремума. Несмотря на то, что производная в этой точке равна нулю, функция убывает как до, так и после этой точки. Это точка перегиба.
Ответ: Точки локального максимума: -3; 1; 6. Точки локального минимума: -5; -1; 4.
в) Решите неравенство f'(x) > 0, f'(x) ≥ 0, f'(x) < 0, f'(x) ≤ 0.
Знак производной функции $f'(x)$ напрямую связан с поведением самой функции $f(x)$:
– если $f'(x) > 0$, функция $f(x)$ строго возрастает;
– если $f'(x) < 0$, функция $f(x)$ строго убывает.
Решим каждое неравенство, анализируя график функции на области определения $[-7; 7]$.
Решение для $f'(x) > 0$:
Неравенство выполняется на интервалах, где функция $f(x)$ строго возрастает.
По графику это интервалы $(-5, -3)$, $(-1, 1)$ и $(4, 6)$.
Решение для $f'(x) \ge 0$:
Неравенство выполняется на промежутках, где функция $f(x)$ не убывает (т.е. возрастает или имеет горизонтальную касательную). Для этого к интервалам строгого возрастания нужно добавить точки, где $f'(x)=0$.
Получаем отрезки $[-5, -3]$, $[-1, 1]$ и $[4, 6]$. Также необходимо учесть изолированную точку $x=2$, где $f'(2)=0$, которая тоже удовлетворяет неравенству.
Решение для $f'(x) < 0$:
Неравенство выполняется на интервалах, где функция $f(x)$ строго убывает.
По графику это интервалы $(-7, -5)$, $(-3, -1)$, а также $(1, 4)$ и $(6, 7)$. Однако в точке $x=2$ производная равна нулю, поэтому эту точку нужно исключить. Таким образом, получаем объединение интервалов: $(-7, -5) \cup (-3, -1) \cup (1, 2) \cup (2, 4) \cup (6, 7)$.
Решение для $f'(x) \le 0$:
Неравенство выполняется на промежутках, где функция $f(x)$ не возрастает (т.е. убывает или имеет горизонтальную касательную). Для этого к интервалам строгого убывания нужно добавить точки, где $f'(x)=0$.
Получаем отрезки $[-7, -5]$, $[-3, -1]$, $[1, 4]$ (этот отрезок включает точку $x=2$, где $f'(2)=0$) и $[6, 7]$.
Ответ:
$f'(x) > 0$ при $x \in (-5, -3) \cup (-1, 1) \cup (4, 6)$.
$f'(x) \ge 0$ при $x \in [-5, -3] \cup [-1, 1] \cup [4, 6] \cup \{2\}$.
$f'(x) < 0$ при $x \in (-7, -5) \cup (-3, -1) \cup (1, 2) \cup (2, 4) \cup (6, 7)$.
$f'(x) \le 0$ при $x \in [-7, -5] \cup [-3, -1] \cup [1, 4] \cup [6, 7]$.
Другие задания:
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