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Номер 11, страница 127, 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 5. Исследование функций и построение графиков. 5.2. Исследование функций и построение графиков - номер 11, страница 127.
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" => 1120434 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "127" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-5-2zad-11" "field_display_title" => "11" "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 {#1046 #items: array:1 [ 0 => App\Models\Branch {#1045 #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 {#1050 #items: array:1 [ 0 => App\Models\Term {#1047 #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 {#1079 #items: array:1 [ 0 => App\Models\Book {#1049 #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 {#1051 #items: array:1 [ 0 => App\Models\Term {#1048 #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 {#1052 #items: array:1 [ 0 => App\Models\Term {#1053 #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 {#1054 #items: array:1 [ 0 => App\Models\Term {#1055 #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 {#1056 #items: array:3 [ 0 => App\Models\Term {#1057 #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 {#1058 #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 {#1059 #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 {#1060 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1061 #items: array:1 [ 0 => App\Models\Term {#1062 #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 {#1063 #items: array:1 [ 0 => App\Models\Term {#1064 #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 {#1065 #items: array:1 [ 0 => App\Models\Term {#1066 #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 {#1067 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1070 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1072 #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 {#1073 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1074 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1075 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1076 #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 {#1051} "field_class" => Illuminate\Database\Eloquent\Collection {#1052} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1054} "field_author" => Illuminate\Database\Eloquent\Collection {#1056} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1060} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1061} "field_country" => Illuminate\Database\Eloquent\Collection {#1063} "field_city" => Illuminate\Database\Eloquent\Collection {#1065} "field_series" => Illuminate\Database\Eloquent\Collection {#1067} "field_umk" => Illuminate\Database\Eloquent\Collection {#1068} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1069} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1070} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1071} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1072} "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 {#1073} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1074} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1075} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1076} "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 {#1077 #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 {#1050} "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 {#1079} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1077} ] #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 {#1107 #items: array:4 [ 0 => App\Models\Branch {#1117 #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" 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#touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1121 #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 {#1125 #items: array:1 [ 0 => App\Models\Term {#1122 #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 {#1124 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1123 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #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 {#1125} "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 {#1124} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1123} ] #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 {#1126 #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" => 1119453 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Исследование функций и построение графиков" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1130 #items: array:1 [ 0 => App\Models\Term {#1127 #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" => "114" "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 {#1129 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1138 #items: array:1 [ 0 => App\Models\Branch {#1137 #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 {#1139 …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 {#1140 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1141 …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 {#1139 …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 {#1140 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1141 …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" => 1119453 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Исследование функций и построение графиков" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1130} "field_page_start" => "114" "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 {#1129} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1138} ] #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 {#1131 #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" => 1119454 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "5.2. Исследование функций и построение графиков" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1128 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "117" "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 {#1134 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1147 #items: array:1 [ 0 => App\Models\Branch {#1146 #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" => 1119453 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Исследование функций и построение графиков" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1148 …2} "field_page_start" => "114" "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 {#1149 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1150 …2} ] #original: array:24 [ "id" => 1119453 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Исследование функций и построение графиков" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1148 …2} "field_page_start" => "114" "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 {#1149 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1150 …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" => 1119454 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "5.2. Исследование функций и построение графиков" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1128} "field_page_start" => "117" "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 {#1134} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1147} ] #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 {#1038 #items: array:2 [ 0 => App\Models\Element {#1098 #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" => 1301119 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1088 #items: array:1 [ 0 => App\Models\Edition {#1097 #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 {#1095 …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 {#1094 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1093 …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 {#1095 …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 {#1094 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1093 …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" => "1120434" "type" => "task" ] "text" => "<p><strong>11. (3)</strong> а) Исследуйте функцию $y=f(x)$ и постройте ее график, где $f(x)=-x^3+2x^2-x$.</p><p><strong>б)</strong> Используя построенный график, определите число корней уравнения $f(x)=g(x)$, где $g(x)=x+5$.</p><p><strong>в)</strong> Определите число корней уравнения $-x^3+2x^2-x=a$ в зависимости от значений параметра $a$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "11-1.jpg" "alt" => null "width" => "1677" "height" => 503 "path" => "/media/algebra_10/baspasy-u19/0-6-5-2zad/11-1.webp?ts=1753534949" ] ] ] #original: array:7 [ "id" => 1301119 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1088} "task" => array:2 [ "refs" => "1120434" "type" => "task" ] "text" => "<p><strong>11. (3)</strong> а) Исследуйте функцию $y=f(x)$ и постройте ее график, где $f(x)=-x^3+2x^2-x$.</p><p><strong>б)</strong> Используя построенный график, определите число корней уравнения $f(x)=g(x)$, где $g(x)=x+5$.</p><p><strong>в)</strong> Определите число корней уравнения $-x^3+2x^2-x=a$ в зависимости от значений параметра $a$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "11-1.jpg" "alt" => null "width" => "1677" "height" => 503 "path" => "/media/algebra_10/baspasy-u19/0-6-5-2zad/11-1.webp?ts=1753534949" ] ] ] #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 {#1090 #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" => 1552155 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1082 #items: array:1 [ 0 => App\Models\Edition {#1091 #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 {#1087 …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 {#1083 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1085 …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 {#1087 …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 {#1083 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1085 …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" => "1120434" "type" => "task" ] "text" => "<p><strong>а)</strong> Исследуем функцию $f(x)=-x^3+2x^2-x$ и построим ее график.</p><p>1. <span>Область определения:</span> Функция является многочленом, поэтому ее область определения — все действительные числа, $D(f)=(-\infty; +\infty)$.</p><p>2. <span>Точки пересечения с осями координат:</span></p><p> - <span>С осью Oy (при $x=0$):</span> $f(0)=-0^3+2 \cdot 0^2 - 0 = 0$. Точка пересечения — $(0; 0)$.</p><p> - <span>С осью Ox (при $f(x)=0$):</span> $-x^3+2x^2-x=0 \implies -x(x^2-2x+1)=0 \implies -x(x-1)^2=0$. Корни уравнения: $x_1=0$ и $x_2=1$ (корень кратности 2). Точки пересечения — $(0; 0)$ и $(1; 0)$.</p><p>3. <span>Четность и периодичность:</span></p><p> $f(-x) = -(-x)^3+2(-x)^2-(-x) = x^3+2x^2+x$. Так как $f(-x) \neq f(x)$ и $f(-x) \neq -f(x)$, функция является функцией общего вида (ни четной, ни нечетной). Функция непериодическая.</p><p>4. <span>Промежутки монотонности и точки экстремума:</span></p><p> Найдем первую производную функции: $f'(x)=(-x^3+2x^2-x)' = -3x^2+4x-1$.</p><p> Найдем критические точки, решив уравнение $f'(x)=0$: $-3x^2+4x-1=0$, или $3x^2-4x+1=0$. Корни уравнения: $x_1=1/3$ и $x_2=1$.</p><p> Исследуем знак производной на интервалах, на которые критические точки делят область определения:</p><p> - <span>На интервале $(-\infty; 1/3)$</span>, $f'(x)<0$, следовательно, функция убывает.</p><p> - <span>На интервале $(1/3; 1)$</span>, $f'(x)>0$, следовательно, функция возрастает.</p><p> - <span>На интервале $(1; +\infty)$</span>, $f'(x)<0$, следовательно, функция убывает.</p><p> В точке $x=1/3$ производная меняет знак с «–» на «+», значит, это точка локального минимума. Значение функции в этой точке: $f_{min} = f(1/3) = -(1/3)^3+2(1/3)^2-1/3 = -1/27+2/9-1/3 = -4/27$.</p><p> В точке $x=1$ производная меняет знак с «+» на «–», значит, это точка локального максимума. Значение функции в этой точке: $f_{max} = f(1) = -1^3+2(1)^2-1 = 0$.</p><p>5. <span>Промежутки выпуклости и точки перегиба:</span></p><p> Найдем вторую производную: $f''(x)=(-3x^2+4x-1)' = -6x+4$.</p><p> Найдем точки, в которых $f''(x)=0$: $-6x+4=0 \implies x=2/3$.</p><p> - <span>При $x < 2/3$</span>, $f''(x)>0$, график функции вогнутый (выпуклый вниз).</p><p> - <span>При $x > 2/3$</span>, $f''(x)<0$, график функции выпуклый (выпуклый вверх).</p><p> Так как в точке $x=2/3$ меняется направление выпуклости, это точка перегиба. Значение функции в этой точке: $f(2/3) = -(2/3)^3+2(2/3)^2-2/3 = -8/27+8/9-2/3 = -2/27$.</p><p>6. <span>Построение графика:</span></p><p> На основе полученных данных, график функции представляет собой кривую, которая приходит из $+\infty$ (при $x \to -\infty$), проходит через точку $(0;0)$, убывает до точки локального минимума $(1/3; -4/27)$, затем возрастает, меняя в точке перегиба $(2/3; -2/27)$ выпуклость вниз на выпуклость вверх, достигает точки локального максимума $(1;0)$, где касается оси Ox, и после этого убывает в $-\infty$ (при $x \to +\infty$).</p><p><strong>Ответ:</strong> Функция исследована. Ее график — кубическая парабола с локальным максимумом в точке $(1; 0)$ и локальным минимумом в точке $(1/3; -4/27)$.</p><p><strong>б)</strong> Число корней уравнения $f(x)=g(x)$ соответствует числу точек пересечения графиков функций $y=f(x)$ и $y=g(x)$, где $g(x)=x+5$.</p><p>График $y=f(x)$ был исследован в пункте а). График $y=g(x)=x+5$ — это прямая линия, проходящая через точки $(-5; 0)$ и $(0; 5)$.</p><p>Чтобы найти число точек пересечения аналитически, рассмотрим уравнение $f(x)-g(x)=0$.</p><p>$-x^3+2x^2-x - (x+5) = 0$</p><p>$-x^3+2x^2-2x-5 = 0$</p><p>$x^3-2x^2+2x+5 = 0$</p><p>Пусть $h(x) = x^3-2x^2+2x+5$. Исследуем эту функцию на монотонность. Найдем ее производную:</p><p>$h'(x) = 3x^2-4x+2$.</p><p>Найдем дискриминант квадратного трехчлена $3x^2-4x+2$: $D = (-4)^2 - 4 \cdot 3 \cdot 2 = 16-24=-8$.</p><p>Поскольку дискриминант $D<0$ и старший коэффициент $3>0$, то $h'(x)>0$ при всех значениях $x$. Это означает, что функция $h(x)$ является строго возрастающей на всей числовой оси.</p><p>Строго монотонная функция может пересекать ось Ox (т.е. принимать значение 0) не более одного раза. Так как $\lim_{x\to-\infty} h(x) = -\infty$ и $\lim_{x\to+\infty} h(x) = +\infty$, функция $h(x)$ пересекает ось Ox ровно один раз. Следовательно, уравнение $h(x)=0$ имеет единственный корень.</p><p><strong>Ответ:</strong> 1 корень.</p><p><strong>в)</strong> Требуется определить число корней уравнения $-x^3+2x^2-x=a$ в зависимости от параметра $a$. Это равносильно нахождению числа точек пересечения графика функции $y=f(x)=-x^3+2x^2-x$ с горизонтальной прямой $y=a$.</p><p>В пункте а) мы нашли экстремумы функции $f(x)$:</p><p>- <span>Локальный максимум:</span> $y_{max} = f(1) = 0$.</p><p>- <span>Локальный минимум:</span> $y_{min} = f(1/3) = -4/27$.</p><p>Теперь рассмотрим, сколько раз прямая $y=a$ пересекает график $y=f(x)$ при различных значениях $a$:</p><p>- <span>Если $a > y_{max}$ (т.е. $a > 0$)</span>, прямая $y=a$ находится выше локального максимума и пересекает график в одной точке. Уравнение имеет 1 корень.</p><p>- <span>Если $a = y_{max}$ (т.е. $a = 0$)</span>, прямая $y=a$ касается графика в точке максимума и пересекает его в другой точке. Уравнение имеет 2 корня.</p><p>- <span>Если $y_{min} < a < y_{max}$ (т.е. $-4/27 < a < 0$)</span>, прямая $y=a$ проходит между экстремумами и пересекает график в трех точках. Уравнение имеет 3 корня.</p><p>- <span>Если $a = y_{min}$ (т.е. $a = -4/27$)</span>, прямая $y=a$ касается графика в точке минимума и пересекает его в другой точке. Уравнение имеет 2 корня.</p><p>- <span>Если $a < y_{min}$ (т.е. $a < -4/27$)</span>, прямая $y=a$ находится ниже локального минимума и пересекает график в одной точке. Уравнение имеет 1 корень.</p><p><strong>Ответ:</strong> </p><p>- при $a \in (-\infty; -4/27) \cup (0; +\infty)$ — 1 корень;</p><p>- при $a = -4/27$ и $a=0$ — 2 корня;</p><p>- при $a \in (-4/27; 0)$ — 3 корня.</p>" ] #original: array:6 [ "id" => 1552155 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1082} "task" => array:2 [ "refs" => "1120434" "type" => "task" ] "text" => "<p><strong>а)</strong> Исследуем функцию $f(x)=-x^3+2x^2-x$ и построим ее график.</p><p>1. <span>Область определения:</span> Функция является многочленом, поэтому ее область определения — все действительные числа, $D(f)=(-\infty; +\infty)$.</p><p>2. <span>Точки пересечения с осями координат:</span></p><p> - <span>С осью Oy (при $x=0$):</span> $f(0)=-0^3+2 \cdot 0^2 - 0 = 0$. Точка пересечения — $(0; 0)$.</p><p> - <span>С осью Ox (при $f(x)=0$):</span> $-x^3+2x^2-x=0 \implies -x(x^2-2x+1)=0 \implies -x(x-1)^2=0$. Корни уравнения: $x_1=0$ и $x_2=1$ (корень кратности 2). Точки пересечения — $(0; 0)$ и $(1; 0)$.</p><p>3. <span>Четность и периодичность:</span></p><p> $f(-x) = -(-x)^3+2(-x)^2-(-x) = x^3+2x^2+x$. Так как $f(-x) \neq f(x)$ и $f(-x) \neq -f(x)$, функция является функцией общего вида (ни четной, ни нечетной). Функция непериодическая.</p><p>4. <span>Промежутки монотонности и точки экстремума:</span></p><p> Найдем первую производную функции: $f'(x)=(-x^3+2x^2-x)' = -3x^2+4x-1$.</p><p> Найдем критические точки, решив уравнение $f'(x)=0$: $-3x^2+4x-1=0$, или $3x^2-4x+1=0$. Корни уравнения: $x_1=1/3$ и $x_2=1$.</p><p> Исследуем знак производной на интервалах, на которые критические точки делят область определения:</p><p> - <span>На интервале $(-\infty; 1/3)$</span>, $f'(x)<0$, следовательно, функция убывает.</p><p> - <span>На интервале $(1/3; 1)$</span>, $f'(x)>0$, следовательно, функция возрастает.</p><p> - <span>На интервале $(1; +\infty)$</span>, $f'(x)<0$, следовательно, функция убывает.</p><p> В точке $x=1/3$ производная меняет знак с «–» на «+», значит, это точка локального минимума. Значение функции в этой точке: $f_{min} = f(1/3) = -(1/3)^3+2(1/3)^2-1/3 = -1/27+2/9-1/3 = -4/27$.</p><p> В точке $x=1$ производная меняет знак с «+» на «–», значит, это точка локального максимума. Значение функции в этой точке: $f_{max} = f(1) = -1^3+2(1)^2-1 = 0$.</p><p>5. <span>Промежутки выпуклости и точки перегиба:</span></p><p> Найдем вторую производную: $f''(x)=(-3x^2+4x-1)' = -6x+4$.</p><p> Найдем точки, в которых $f''(x)=0$: $-6x+4=0 \implies x=2/3$.</p><p> - <span>При $x < 2/3$</span>, $f''(x)>0$, график функции вогнутый (выпуклый вниз).</p><p> - <span>При $x > 2/3$</span>, $f''(x)<0$, график функции выпуклый (выпуклый вверх).</p><p> Так как в точке $x=2/3$ меняется направление выпуклости, это точка перегиба. Значение функции в этой точке: $f(2/3) = -(2/3)^3+2(2/3)^2-2/3 = -8/27+8/9-2/3 = -2/27$.</p><p>6. <span>Построение графика:</span></p><p> На основе полученных данных, график функции представляет собой кривую, которая приходит из $+\infty$ (при $x \to -\infty$), проходит через точку $(0;0)$, убывает до точки локального минимума $(1/3; -4/27)$, затем возрастает, меняя в точке перегиба $(2/3; -2/27)$ выпуклость вниз на выпуклость вверх, достигает точки локального максимума $(1;0)$, где касается оси Ox, и после этого убывает в $-\infty$ (при $x \to +\infty$).</p><p><strong>Ответ:</strong> Функция исследована. Ее график — кубическая парабола с локальным максимумом в точке $(1; 0)$ и локальным минимумом в точке $(1/3; -4/27)$.</p><p><strong>б)</strong> Число корней уравнения $f(x)=g(x)$ соответствует числу точек пересечения графиков функций $y=f(x)$ и $y=g(x)$, где $g(x)=x+5$.</p><p>График $y=f(x)$ был исследован в пункте а). График $y=g(x)=x+5$ — это прямая линия, проходящая через точки $(-5; 0)$ и $(0; 5)$.</p><p>Чтобы найти число точек пересечения аналитически, рассмотрим уравнение $f(x)-g(x)=0$.</p><p>$-x^3+2x^2-x - (x+5) = 0$</p><p>$-x^3+2x^2-2x-5 = 0$</p><p>$x^3-2x^2+2x+5 = 0$</p><p>Пусть $h(x) = x^3-2x^2+2x+5$. Исследуем эту функцию на монотонность. Найдем ее производную:</p><p>$h'(x) = 3x^2-4x+2$.</p><p>Найдем дискриминант квадратного трехчлена $3x^2-4x+2$: $D = (-4)^2 - 4 \cdot 3 \cdot 2 = 16-24=-8$.</p><p>Поскольку дискриминант $D<0$ и старший коэффициент $3>0$, то $h'(x)>0$ при всех значениях $x$. Это означает, что функция $h(x)$ является строго возрастающей на всей числовой оси.</p><p>Строго монотонная функция может пересекать ось Ox (т.е. принимать значение 0) не более одного раза. Так как $\lim_{x\to-\infty} h(x) = -\infty$ и $\lim_{x\to+\infty} h(x) = +\infty$, функция $h(x)$ пересекает ось Ox ровно один раз. Следовательно, уравнение $h(x)=0$ имеет единственный корень.</p><p><strong>Ответ:</strong> 1 корень.</p><p><strong>в)</strong> Требуется определить число корней уравнения $-x^3+2x^2-x=a$ в зависимости от параметра $a$. Это равносильно нахождению числа точек пересечения графика функции $y=f(x)=-x^3+2x^2-x$ с горизонтальной прямой $y=a$.</p><p>В пункте а) мы нашли экстремумы функции $f(x)$:</p><p>- <span>Локальный максимум:</span> $y_{max} = f(1) = 0$.</p><p>- <span>Локальный минимум:</span> $y_{min} = f(1/3) = -4/27$.</p><p>Теперь рассмотрим, сколько раз прямая $y=a$ пересекает график $y=f(x)$ при различных значениях $a$:</p><p>- <span>Если $a > y_{max}$ (т.е. $a > 0$)</span>, прямая $y=a$ находится выше локального максимума и пересекает график в одной точке. Уравнение имеет 1 корень.</p><p>- <span>Если $a = y_{max}$ (т.е. $a = 0$)</span>, прямая $y=a$ касается графика в точке максимума и пересекает его в другой точке. Уравнение имеет 2 корня.</p><p>- <span>Если $y_{min} < a < y_{max}$ (т.е. $-4/27 < a < 0$)</span>, прямая $y=a$ проходит между экстремумами и пересекает график в трех точках. Уравнение имеет 3 корня.</p><p>- <span>Если $a = y_{min}$ (т.е. $a = -4/27$)</span>, прямая $y=a$ касается графика в точке минимума и пересекает его в другой точке. Уравнение имеет 2 корня.</p><p>- <span>Если $a < y_{min}$ (т.е. $a < -4/27$)</span>, прямая $y=a$ находится ниже локального минимума и пересекает график в одной точке. Уравнение имеет 1 корень.</p><p><strong>Ответ:</strong> </p><p>- при $a \in (-\infty; -4/27) \cup (0; +\infty)$ — 1 корень;</p><p>- при $a = -4/27$ и $a=0$ — 2 корня;</p><p>- при $a \in (-4/27; 0)$ — 3 корня.</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" => "1120435" "type" => "task" ] "previous" => array:2 [ "refs" => "1120433" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1101 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1040 #items: array:1 [ 0 => App\Models\BookPage {#1102 #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" => 1099280 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "127" "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/page-2-127" "field_display_title" => "127" "field_folder" => "2" "field_image_name" => "127" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1084 #items: array:1 [ 0 => App\Models\Branch {#1117} ] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1081 #items: array:1 [ 0 => App\Models\Book {#1049} ] #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 {#1078 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1114 #items: array:1 [ 0 => App\Models\Element {#1112 #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" => 1262705 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1110 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1262705 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1110 …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" => "1099281" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1099279" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1220 #items: array:8 [ 0 => App\Models\Task {#1234 #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" => 1120429 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "126" "field_page_end" => "127" "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-5-2zad-6" "field_display_title" => "6" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1235 …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 {#1236 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1237 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1238 …2} "content" => Illuminate\Database\Eloquent\Collection {#1240 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1246 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1120429 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "126" "field_page_end" => "127" "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-5-2zad-6" "field_display_title" => "6" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1235 …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 {#1236 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1237 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1238 …2} "content" => Illuminate\Database\Eloquent\Collection {#1240 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1246 …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 {#1241 #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" => 1120430 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "127" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-5-2zad-7" "field_display_title" => "7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1239 …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 {#1248 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1244 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1242 …2} "content" => Illuminate\Database\Eloquent\Collection {#1243 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1258 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1120430 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "127" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-5-2zad-7" "field_display_title" => "7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1239 …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 {#1248 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1244 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1242 …2} "content" => Illuminate\Database\Eloquent\Collection {#1243 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1258 …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 {#1253 #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" => 1120431 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "127" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-5-2zad-8" "field_display_title" => "8" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1245 …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 {#1260 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1256 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1254 …2} "content" => Illuminate\Database\Eloquent\Collection {#1255 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1270 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1120431 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "127" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-5-2zad-8" "field_display_title" => "8" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1245 …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 {#1260 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1256 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1254 …2} "content" => Illuminate\Database\Eloquent\Collection {#1255 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1270 …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 {#1265 #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" => 1120432 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "127" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-5-2zad-9" "field_display_title" => "9" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1257 …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 {#1272 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1268 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1266 …2} "content" => Illuminate\Database\Eloquent\Collection {#1267 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1282 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1120432 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "127" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-5-2zad-9" "field_display_title" => "9" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1257 …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 {#1272 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1268 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1266 …2} "content" => Illuminate\Database\Eloquent\Collection {#1267 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1282 …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 {#1277 #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" => 1120433 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "127" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-5-2zad-10" "field_display_title" => "10" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1269 …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 {#1284 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1280 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1278 …2} "content" => Illuminate\Database\Eloquent\Collection {#1279 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1294 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1120433 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "127" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-5-2zad-10" "field_display_title" => "10" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1269 …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 {#1284 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1280 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1278 …2} "content" => Illuminate\Database\Eloquent\Collection {#1279 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1294 …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 => "*" ] } 5 => App\Models\Task {#1289 #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" => 1120434 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "127" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-5-2zad-11" "field_display_title" => "11" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1281 …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 {#1296 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1292 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1290 …2} "content" => Illuminate\Database\Eloquent\Collection {#1293 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => 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№11 (с. 127)
Условие. №11 (с. 127)
скриншот условия
11. (3) а) Исследуйте функцию $y=f(x)$ и постройте ее график, где $f(x)=-x^3+2x^2-x$.
б) Используя построенный график, определите число корней уравнения $f(x)=g(x)$, где $g(x)=x+5$.
в) Определите число корней уравнения $-x^3+2x^2-x=a$ в зависимости от значений параметра $a$.
Решение 2 (rus). №11 (с. 127)
а) Исследуем функцию $f(x)=-x^3+2x^2-x$ и построим ее график.
1. Область определения: Функция является многочленом, поэтому ее область определения — все действительные числа, $D(f)=(-\infty; +\infty)$.
2. Точки пересечения с осями координат:
- С осью Oy (при $x=0$): $f(0)=-0^3+2 \cdot 0^2 - 0 = 0$. Точка пересечения — $(0; 0)$.
- С осью Ox (при $f(x)=0$): $-x^3+2x^2-x=0 \implies -x(x^2-2x+1)=0 \implies -x(x-1)^2=0$. Корни уравнения: $x_1=0$ и $x_2=1$ (корень кратности 2). Точки пересечения — $(0; 0)$ и $(1; 0)$.
3. Четность и периодичность:
$f(-x) = -(-x)^3+2(-x)^2-(-x) = x^3+2x^2+x$. Так как $f(-x) \neq f(x)$ и $f(-x) \neq -f(x)$, функция является функцией общего вида (ни четной, ни нечетной). Функция непериодическая.
4. Промежутки монотонности и точки экстремума:
Найдем первую производную функции: $f'(x)=(-x^3+2x^2-x)' = -3x^2+4x-1$.
Найдем критические точки, решив уравнение $f'(x)=0$: $-3x^2+4x-1=0$, или $3x^2-4x+1=0$. Корни уравнения: $x_1=1/3$ и $x_2=1$.
Исследуем знак производной на интервалах, на которые критические точки делят область определения:
- На интервале $(-\infty; 1/3)$, $f'(x)<0$, следовательно, функция убывает.
- На интервале $(1/3; 1)$, $f'(x)>0$, следовательно, функция возрастает.
- На интервале $(1; +\infty)$, $f'(x)<0$, следовательно, функция убывает.
В точке $x=1/3$ производная меняет знак с «–» на «+», значит, это точка локального минимума. Значение функции в этой точке: $f_{min} = f(1/3) = -(1/3)^3+2(1/3)^2-1/3 = -1/27+2/9-1/3 = -4/27$.
В точке $x=1$ производная меняет знак с «+» на «–», значит, это точка локального максимума. Значение функции в этой точке: $f_{max} = f(1) = -1^3+2(1)^2-1 = 0$.
5. Промежутки выпуклости и точки перегиба:
Найдем вторую производную: $f''(x)=(-3x^2+4x-1)' = -6x+4$.
Найдем точки, в которых $f''(x)=0$: $-6x+4=0 \implies x=2/3$.
- При $x < 2/3$, $f''(x)>0$, график функции вогнутый (выпуклый вниз).
- При $x > 2/3$, $f''(x)<0$, график функции выпуклый (выпуклый вверх).
Так как в точке $x=2/3$ меняется направление выпуклости, это точка перегиба. Значение функции в этой точке: $f(2/3) = -(2/3)^3+2(2/3)^2-2/3 = -8/27+8/9-2/3 = -2/27$.
6. Построение графика:
На основе полученных данных, график функции представляет собой кривую, которая приходит из $+\infty$ (при $x \to -\infty$), проходит через точку $(0;0)$, убывает до точки локального минимума $(1/3; -4/27)$, затем возрастает, меняя в точке перегиба $(2/3; -2/27)$ выпуклость вниз на выпуклость вверх, достигает точки локального максимума $(1;0)$, где касается оси Ox, и после этого убывает в $-\infty$ (при $x \to +\infty$).
Ответ: Функция исследована. Ее график — кубическая парабола с локальным максимумом в точке $(1; 0)$ и локальным минимумом в точке $(1/3; -4/27)$.
б) Число корней уравнения $f(x)=g(x)$ соответствует числу точек пересечения графиков функций $y=f(x)$ и $y=g(x)$, где $g(x)=x+5$.
График $y=f(x)$ был исследован в пункте а). График $y=g(x)=x+5$ — это прямая линия, проходящая через точки $(-5; 0)$ и $(0; 5)$.
Чтобы найти число точек пересечения аналитически, рассмотрим уравнение $f(x)-g(x)=0$.
$-x^3+2x^2-x - (x+5) = 0$
$-x^3+2x^2-2x-5 = 0$
$x^3-2x^2+2x+5 = 0$
Пусть $h(x) = x^3-2x^2+2x+5$. Исследуем эту функцию на монотонность. Найдем ее производную:
$h'(x) = 3x^2-4x+2$.
Найдем дискриминант квадратного трехчлена $3x^2-4x+2$: $D = (-4)^2 - 4 \cdot 3 \cdot 2 = 16-24=-8$.
Поскольку дискриминант $D<0$ и старший коэффициент $3>0$, то $h'(x)>0$ при всех значениях $x$. Это означает, что функция $h(x)$ является строго возрастающей на всей числовой оси.
Строго монотонная функция может пересекать ось Ox (т.е. принимать значение 0) не более одного раза. Так как $\lim_{x\to-\infty} h(x) = -\infty$ и $\lim_{x\to+\infty} h(x) = +\infty$, функция $h(x)$ пересекает ось Ox ровно один раз. Следовательно, уравнение $h(x)=0$ имеет единственный корень.
Ответ: 1 корень.
в) Требуется определить число корней уравнения $-x^3+2x^2-x=a$ в зависимости от параметра $a$. Это равносильно нахождению числа точек пересечения графика функции $y=f(x)=-x^3+2x^2-x$ с горизонтальной прямой $y=a$.
В пункте а) мы нашли экстремумы функции $f(x)$:
- Локальный максимум: $y_{max} = f(1) = 0$.
- Локальный минимум: $y_{min} = f(1/3) = -4/27$.
Теперь рассмотрим, сколько раз прямая $y=a$ пересекает график $y=f(x)$ при различных значениях $a$:
- Если $a > y_{max}$ (т.е. $a > 0$), прямая $y=a$ находится выше локального максимума и пересекает график в одной точке. Уравнение имеет 1 корень.
- Если $a = y_{max}$ (т.е. $a = 0$), прямая $y=a$ касается графика в точке максимума и пересекает его в другой точке. Уравнение имеет 2 корня.
- Если $y_{min} < a < y_{max}$ (т.е. $-4/27 < a < 0$), прямая $y=a$ проходит между экстремумами и пересекает график в трех точках. Уравнение имеет 3 корня.
- Если $a = y_{min}$ (т.е. $a = -4/27$), прямая $y=a$ касается графика в точке минимума и пересекает его в другой точке. Уравнение имеет 2 корня.
- Если $a < y_{min}$ (т.е. $a < -4/27$), прямая $y=a$ находится ниже локального минимума и пересекает график в одной точке. Уравнение имеет 1 корень.
Ответ:
- при $a \in (-\infty; -4/27) \cup (0; +\infty)$ — 1 корень;
- при $a = -4/27$ и $a=0$ — 2 корня;
- при $a \in (-4/27; 0)$ — 3 корня.
Другие задания:
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