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Номер 22.10, страница 121 - гдз по алгебре 10 класс учебник Абылкасымова, Жумагулова
Авторы: Абылкасымова А. Е., Жумагулова З. А.
Тип: Учебник
Издательство: Мектеп
Год издания: 2019 - 2026
ISBN: 978-601-07-1142-6
Утверждено Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 10 классе
Глава 6. Применение производной. Параграф 22. Наибольшее и наименьшее значения функции на отрезке - номер 22.10, страница 121.
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" => 1113970 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/22-10" "field_display_title" => "22.10" "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" => 1113580 "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 {#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" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "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 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "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" => "106" "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 {#1078 #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" => 4293 "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 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "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 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "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 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "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 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "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" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #original: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #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:2 [ 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" => 6997 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абылкасымова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алма" "field_patronymic" => "Есимбековна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6997 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абылкасымова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алма" "field_patronymic" => "Есимбековна" "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" => 6999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумагулова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Зауре" "field_patronymic" => "Абдыкеновна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумагулова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Зауре" "field_patronymic" => "Абдыкеновна" "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 {#1059 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1060 #items: array:1 [ 0 => App\Models\Term {#1061 #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 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" …1 ] "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 {#1062 #items: array:1 [ 0 => App\Models\Term {#1063 #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 {#1064 #items: array:1 [ 0 => App\Models\Term {#1065 #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 {#1066 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1070 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 10 класса общественно-гуманитарного направления общеобразовательных школ" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "514" "field_priority" => "2" "field_default_folder" => "/algebra_10/Abylkasymova-u/" "field_isbn" => "978-601-07-1142-6" "field_cover" => array:1 [ 0 => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1072 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1073 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1074 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/abylkasymova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1075 #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" => 4293 "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 {#1059} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1060} "field_country" => Illuminate\Database\Eloquent\Collection {#1062} "field_city" => Illuminate\Database\Eloquent\Collection {#1064} "field_series" => Illuminate\Database\Eloquent\Collection {#1066} "field_umk" => Illuminate\Database\Eloquent\Collection {#1067} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1068} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1069} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1070} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1071} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 10 класса общественно-гуманитарного направления общеобразовательных школ" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "514" "field_priority" => "2" "field_default_folder" => "/algebra_10/Abylkasymova-u/" "field_isbn" => "978-601-07-1142-6" "field_cover" => array:1 [ 0 => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1072} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1073} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1074} "field_url" => "/10-klass/algebra/abylkasymova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1075} "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 {#1076 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1113580 "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 {#1050} "field_page_start" => "106" "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 {#1078} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1076} ] #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 {#1105 #items: array:2 [ 0 => App\Models\Branch {#1113 #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" => 1113580 "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 {#1114 #items: array:1 [ 0 => App\Models\Term {#1047} ] #escapeWhenCastingToString: false } "field_page_start" => "106" "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 {#1115 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1116 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1113580 "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 {#1114} "field_page_start" => "106" "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 {#1115} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1116} ] #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 {#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" => 1113584 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Наибольшее и наименьшее значения функции на отрезке" "field_branch_order" => "22" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1121 #items: array:1 [ 0 => App\Models\Term {#1118 #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 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "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 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "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" => "118" "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 {#1120 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1119 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1113584 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Наибольшее и наименьшее значения функции на отрезке" "field_branch_order" => "22" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1121} "field_page_start" => "118" "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 {#1120} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1119} ] #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 {#1102 #items: array:3 [ 0 => App\Models\Element {#1095 #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" => 1290908 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1087 #items: array:1 [ 0 => App\Models\Edition {#1094 #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" => 4958 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1092 #items: array:1 [ …1] #escapeWhenCastingToString: false } "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 {#1090 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4958 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1092} "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 {#1090} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088} "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" => "1113970" "type" => "task" ] "text" => "<p><strong>22.10. a)</strong> $f(x) = \frac{1}{x} + x^2$, $[0,5; 1];$</p><p><strong>б)</strong> $f(x) = \sqrt{2 - x - x^2}$, $[-1; 0].$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "10-1.jpg" "alt" => null "width" => "2639" "height" => 265 "path" => "/media/algebra_10/Abylkasymova-u/0-22/10-1.webp?ts=1753263150" ] ] ] #original: array:7 [ "id" => 1290908 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1087} "task" => array:2 [ "refs" => "1113970" "type" => "task" ] "text" => "<p><strong>22.10. a)</strong> $f(x) = \frac{1}{x} + x^2$, $[0,5; 1];$</p><p><strong>б)</strong> $f(x) = \sqrt{2 - x - x^2}$, $[-1; 0].$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "10-1.jpg" "alt" => null "width" => "2639" "height" => 265 "path" => "/media/algebra_10/Abylkasymova-u/0-22/10-1.webp?ts=1753263150" ] ] ] #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 {#1089 #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" => 1291422 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1079 #items: array:1 [ 0 => App\Models\Edition {#1086 #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" => 4959 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1082 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1084 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1083 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1080 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4959 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1082} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1084} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1083} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1080} "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" => "1113970" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "10-1.jpg" "alt" => null "width" => "1888" "height" => 2308 "path" => "/media/algebra_10/Abylkasymova-u/1-22/10-1.webp?ts=1753263368" ] 1 => array:5 [ "name" => "10-2.jpg" "alt" => null "width" => "1657" "height" => 1117 "path" => "/media/algebra_10/Abylkasymova-u/1-22/10-2.webp?ts=1753263368" ] ] ] #original: array:6 [ "id" => 1291422 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1079} "task" => array:2 [ "refs" => "1113970" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "10-1.jpg" "alt" => null "width" => "1888" "height" => 2308 "path" => "/media/algebra_10/Abylkasymova-u/1-22/10-1.webp?ts=1753263368" ] 1 => array:5 [ "name" => "10-2.jpg" "alt" => null "width" => "1657" "height" => 1117 "path" => "/media/algebra_10/Abylkasymova-u/1-22/10-2.webp?ts=1753263368" ] ] ] #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\Element {#1081 #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" => 1549551 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ 0 => App\Models\Edition {#1077 #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" => 5655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1041 #items: array:1 [ …1] #escapeWhenCastingToString: false } "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 {#1034 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1039 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1104 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1041} "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 {#1034} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1039} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1104} "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" => "1113970" "type" => "task" ] "text" => "<p><strong>а)</strong> Для нахождения наибольшего и наименьшего значений функции $f(x) = \frac{1}{x} + x^2$ на отрезке $[0.5; 1]$ необходимо найти значения функции на концах отрезка и в критических точках, принадлежащих этому отрезку, а затем сравнить их.</p><p><strong>1.</strong> Найдём производную функции $f(x)$:</p><p>$f'(x) = \left(\frac{1}{x} + x^2\right)' = (x^{-1} + x^2)' = -1 \cdot x^{-2} + 2x = -\frac{1}{x^2} + 2x$.</p><p><strong>2.</strong> Найдём критические точки, приравняв производную к нулю:</p><p>$-\frac{1}{x^2} + 2x = 0$</p><p>$2x = \frac{1}{x^2}$</p><p>$2x^3 = 1$</p><p>$x^3 = \frac{1}{2}$</p><p>$x = \sqrt[3]{\frac{1}{2}} = \frac{1}{\sqrt[3]{2}}$.</p><p><strong>3.</strong> Проверим, принадлежит ли найденная критическая точка $x = \frac{1}{\sqrt[3]{2}}$ отрезку $[0.5; 1]$.</p><p>Поскольку $1^3 = 1$ и $2^3 = 8$, то $1 < \sqrt[3]{2} < 2$. Разделив все части на $\sqrt[3]{2}$ и на 2, получим $\frac{1}{2} < \frac{1}{\sqrt[3]{2}} < 1$.</p><p>Следовательно, $0.5 < \frac{1}{\sqrt[3]{2}} < 1$, и критическая точка принадлежит заданному отрезку.</p><p><strong>4.</strong> Вычислим значения функции в критической точке и на концах отрезка:</p><p>На левом конце отрезка: $f(0.5) = \frac{1}{0.5} + (0.5)^2 = 2 + 0.25 = 2.25$.</p><p>На правом конце отрезка: $f(1) = \frac{1}{1} + 1^2 = 1 + 1 = 2$.</p><p>В критической точке: $f\left(\frac{1}{\sqrt[3]{2}}\right) = \frac{1}{1/\sqrt[3]{2}} + \left(\frac{1}{\sqrt[3]{2}}\right)^2 = \sqrt[3]{2} + \frac{1}{\sqrt[3]{4}} = \frac{\sqrt[3]{2}\sqrt[3]{4}+1}{\sqrt[3]{4}} = \frac{\sqrt[3]{8}+1}{\sqrt[3]{4}} = \frac{3}{\sqrt[3]{4}} = \frac{3\sqrt[3]{2}}{2}$.</p><p><strong>5.</strong> Сравним полученные значения: $2.25$, $2$ и $\frac{3\sqrt[3]{2}}{2}$.</p><p>Чтобы сравнить $\frac{3\sqrt[3]{2}}{2}$ и $2$, сравним $3\sqrt[3]{2}$ и $4$. Возведём обе части в куб: $(3\sqrt[3]{2})^3 = 27 \cdot 2 = 54$ и $4^3 = 64$. Так как $54 < 64$, то $3\sqrt[3]{2} < 4$, и $\frac{3\sqrt[3]{2}}{2} < 2$.</p><p>Таким образом, из трёх значений $2.25$, $2$ и $\frac{3\sqrt[3]{2}}{2}$ наибольшее — это $2.25$, а наименьшее — $\frac{3\sqrt[3]{2}}{2}$.</p><p>Наибольшее значение функции: $\max_{x \in [0.5; 1]} f(x) = f(0.5) = 2.25$.</p><p>Наименьшее значение функции: $\min_{x \in [0.5; 1]} f(x) = f\left(\frac{1}{\sqrt[3]{2}}\right) = \frac{3\sqrt[3]{2}}{2}$.</p><p><strong>Ответ:</strong> наименьшее значение функции равно $\frac{3\sqrt[3]{2}}{2}$, наибольшее значение функции равно $2.25$.</p><p><strong>б)</strong> Для нахождения наибольшего и наименьшего значений функции $f(x) = \sqrt{2 - x - x^2}$ на отрезке $[-1; 0]$ будем использовать тот же алгоритм.</p><p><strong>1.</strong> Сначала найдём область определения функции. Выражение под корнем должно быть неотрицательным: $2 - x - x^2 \ge 0$, или $x^2 + x - 2 \le 0$. Корнями уравнения $x^2 + x - 2 = 0$ являются $x_1 = -2$ и $x_2 = 1$. Неравенство выполняется на отрезке $[-2; 1]$. Заданный отрезок $[-1; 0]$ полностью входит в область определения, поэтому функция на нём непрерывна.</p><p><strong>2.</strong> Найдём производную функции $f(x)$:</p><p>$f'(x) = \left(\sqrt{2 - x - x^2}\right)' = \frac{1}{2\sqrt{2 - x - x^2}} \cdot (2 - x - x^2)' = \frac{-1 - 2x}{2\sqrt{2 - x - x^2}}$.</p><p><strong>3.</strong> Найдём критические точки из условия $f'(x) = 0$:</p><p>$\frac{-1 - 2x}{2\sqrt{2 - x - x^2}} = 0$</p><p>Для этого числитель должен быть равен нулю: $-1 - 2x = 0$, откуда $x = -0.5$.</p><p>Эта точка принадлежит отрезку $[-1; 0]$.</p><p><strong>4.</strong> Вычислим значения функции в критической точке и на концах отрезка:</p><p>$f(-1) = \sqrt{2 - (-1) - (-1)^2} = \sqrt{2 + 1 - 1} = \sqrt{2}$.</p><p>$f(0) = \sqrt{2 - 0 - 0^2} = \sqrt{2}$.</p><p>$f(-0.5) = \sqrt{2 - (-0.5) - (-0.5)^2} = \sqrt{2 + 0.5 - 0.25} = \sqrt{2.25} = 1.5$.</p><p><strong>5.</strong> Сравним полученные значения: $\sqrt{2}$ и $1.5$.</p><p>Так как $1.5 = \sqrt{2.25}$, а $2.25 > 2$, то $1.5 > \sqrt{2}$.</p><p>Наибольшее значение функции: $\max_{x \in [-1; 0]} f(x) = f(-0.5) = 1.5$.</p><p>Наименьшее значение функции: $\min_{x \in [-1; 0]} f(x) = f(-1) = f(0) = \sqrt{2}$.</p><p><strong>Ответ:</strong> наименьшее значение функции равно $\sqrt{2}$, наибольшее значение функции равно $1.5$.</p>" ] #original: array:6 [ "id" => 1549551 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108} "task" => array:2 [ "refs" => "1113970" "type" => "task" ] "text" => "<p><strong>а)</strong> Для нахождения наибольшего и наименьшего значений функции $f(x) = \frac{1}{x} + x^2$ на отрезке $[0.5; 1]$ необходимо найти значения функции на концах отрезка и в критических точках, принадлежащих этому отрезку, а затем сравнить их.</p><p><strong>1.</strong> Найдём производную функции $f(x)$:</p><p>$f'(x) = \left(\frac{1}{x} + x^2\right)' = (x^{-1} + x^2)' = -1 \cdot x^{-2} + 2x = -\frac{1}{x^2} + 2x$.</p><p><strong>2.</strong> Найдём критические точки, приравняв производную к нулю:</p><p>$-\frac{1}{x^2} + 2x = 0$</p><p>$2x = \frac{1}{x^2}$</p><p>$2x^3 = 1$</p><p>$x^3 = \frac{1}{2}$</p><p>$x = \sqrt[3]{\frac{1}{2}} = \frac{1}{\sqrt[3]{2}}$.</p><p><strong>3.</strong> Проверим, принадлежит ли найденная критическая точка $x = \frac{1}{\sqrt[3]{2}}$ отрезку $[0.5; 1]$.</p><p>Поскольку $1^3 = 1$ и $2^3 = 8$, то $1 < \sqrt[3]{2} < 2$. Разделив все части на $\sqrt[3]{2}$ и на 2, получим $\frac{1}{2} < \frac{1}{\sqrt[3]{2}} < 1$.</p><p>Следовательно, $0.5 < \frac{1}{\sqrt[3]{2}} < 1$, и критическая точка принадлежит заданному отрезку.</p><p><strong>4.</strong> Вычислим значения функции в критической точке и на концах отрезка:</p><p>На левом конце отрезка: $f(0.5) = \frac{1}{0.5} + (0.5)^2 = 2 + 0.25 = 2.25$.</p><p>На правом конце отрезка: $f(1) = \frac{1}{1} + 1^2 = 1 + 1 = 2$.</p><p>В критической точке: $f\left(\frac{1}{\sqrt[3]{2}}\right) = \frac{1}{1/\sqrt[3]{2}} + \left(\frac{1}{\sqrt[3]{2}}\right)^2 = \sqrt[3]{2} + \frac{1}{\sqrt[3]{4}} = \frac{\sqrt[3]{2}\sqrt[3]{4}+1}{\sqrt[3]{4}} = \frac{\sqrt[3]{8}+1}{\sqrt[3]{4}} = \frac{3}{\sqrt[3]{4}} = \frac{3\sqrt[3]{2}}{2}$.</p><p><strong>5.</strong> Сравним полученные значения: $2.25$, $2$ и $\frac{3\sqrt[3]{2}}{2}$.</p><p>Чтобы сравнить $\frac{3\sqrt[3]{2}}{2}$ и $2$, сравним $3\sqrt[3]{2}$ и $4$. Возведём обе части в куб: $(3\sqrt[3]{2})^3 = 27 \cdot 2 = 54$ и $4^3 = 64$. Так как $54 < 64$, то $3\sqrt[3]{2} < 4$, и $\frac{3\sqrt[3]{2}}{2} < 2$.</p><p>Таким образом, из трёх значений $2.25$, $2$ и $\frac{3\sqrt[3]{2}}{2}$ наибольшее — это $2.25$, а наименьшее — $\frac{3\sqrt[3]{2}}{2}$.</p><p>Наибольшее значение функции: $\max_{x \in [0.5; 1]} f(x) = f(0.5) = 2.25$.</p><p>Наименьшее значение функции: $\min_{x \in [0.5; 1]} f(x) = f\left(\frac{1}{\sqrt[3]{2}}\right) = \frac{3\sqrt[3]{2}}{2}$.</p><p><strong>Ответ:</strong> наименьшее значение функции равно $\frac{3\sqrt[3]{2}}{2}$, наибольшее значение функции равно $2.25$.</p><p><strong>б)</strong> Для нахождения наибольшего и наименьшего значений функции $f(x) = \sqrt{2 - x - x^2}$ на отрезке $[-1; 0]$ будем использовать тот же алгоритм.</p><p><strong>1.</strong> Сначала найдём область определения функции. Выражение под корнем должно быть неотрицательным: $2 - x - x^2 \ge 0$, или $x^2 + x - 2 \le 0$. Корнями уравнения $x^2 + x - 2 = 0$ являются $x_1 = -2$ и $x_2 = 1$. Неравенство выполняется на отрезке $[-2; 1]$. Заданный отрезок $[-1; 0]$ полностью входит в область определения, поэтому функция на нём непрерывна.</p><p><strong>2.</strong> Найдём производную функции $f(x)$:</p><p>$f'(x) = \left(\sqrt{2 - x - x^2}\right)' = \frac{1}{2\sqrt{2 - x - x^2}} \cdot (2 - x - x^2)' = \frac{-1 - 2x}{2\sqrt{2 - x - x^2}}$.</p><p><strong>3.</strong> Найдём критические точки из условия $f'(x) = 0$:</p><p>$\frac{-1 - 2x}{2\sqrt{2 - x - x^2}} = 0$</p><p>Для этого числитель должен быть равен нулю: $-1 - 2x = 0$, откуда $x = -0.5$.</p><p>Эта точка принадлежит отрезку $[-1; 0]$.</p><p><strong>4.</strong> Вычислим значения функции в критической точке и на концах отрезка:</p><p>$f(-1) = \sqrt{2 - (-1) - (-1)^2} = \sqrt{2 + 1 - 1} = \sqrt{2}$.</p><p>$f(0) = \sqrt{2 - 0 - 0^2} = \sqrt{2}$.</p><p>$f(-0.5) = \sqrt{2 - (-0.5) - (-0.5)^2} = \sqrt{2 + 0.5 - 0.25} = \sqrt{2.25} = 1.5$.</p><p><strong>5.</strong> Сравним полученные значения: $\sqrt{2}$ и $1.5$.</p><p>Так как $1.5 = \sqrt{2.25}$, а $2.25 > 2$, то $1.5 > \sqrt{2}$.</p><p>Наибольшее значение функции: $\max_{x \in [-1; 0]} f(x) = f(-0.5) = 1.5$.</p><p>Наименьшее значение функции: $\min_{x \in [-1; 0]} f(x) = f(-1) = f(0) = \sqrt{2}$.</p><p><strong>Ответ:</strong> наименьшее значение функции равно $\sqrt{2}$, наибольшее значение функции равно $1.5$.</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" => "1113971" "type" => "task" ] "previous" => array:2 [ "refs" => "1113969" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1098 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\BookPage {#1042 #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" => 1098458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/page-121" "field_display_title" => "121" "field_folder" => "1" "field_image_name" => "121" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1040 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1103 #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 {#1112 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1128 #items: array:1 [ 0 => App\Models\Element {#1127 #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" => 1261883 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1129 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1261883 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1129 …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" => "1098459" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1098457" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:10 [ 0 => App\Models\Task {#1214 #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" => 1113967 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/22-7" "field_display_title" => "22.7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1215 …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 {#1216 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1217 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1218 …2} "content" => Illuminate\Database\Eloquent\Collection {#1221 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1227 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113967 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/22-7" "field_display_title" => "22.7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1215 …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 {#1216 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1217 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1218 …2} "content" => Illuminate\Database\Eloquent\Collection {#1221 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1227 …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 {#1220 #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" => 1113968 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/22-8" "field_display_title" => "22.8" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1219 …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 {#1226 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1228 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1224 …2} "content" => Illuminate\Database\Eloquent\Collection {#1223 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1241 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113968 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/22-8" "field_display_title" => "22.8" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1219 …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 {#1226 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1228 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1224 …2} "content" => Illuminate\Database\Eloquent\Collection {#1223 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1241 …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 {#1225 #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" => 1113969 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/22-9" "field_display_title" => "22.9" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1222 …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 {#1240 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1242 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1238 …2} "content" => Illuminate\Database\Eloquent\Collection {#1237 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1255 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113969 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/22-9" "field_display_title" => "22.9" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1222 …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 {#1240 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1242 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1238 …2} "content" => Illuminate\Database\Eloquent\Collection {#1237 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1255 …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 {#1239 #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" => 1113970 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/22-10" "field_display_title" => "22.10" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1236 …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 {#1254 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1256 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1252 …2} "content" => Illuminate\Database\Eloquent\Collection {#1250 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1253 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113970 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/22-10" "field_display_title" => "22.10" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1236 …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 {#1254 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1256 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1252 …2} "content" => Illuminate\Database\Eloquent\Collection {#1250 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1253 …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 {#1251 #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" => 1113971 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/22-11" "field_display_title" => "22.11" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1264 …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 {#1265 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1266 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1267 …2} "content" => Illuminate\Database\Eloquent\Collection {#1270 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1276 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113971 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/22-11" "field_display_title" => "22.11" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1264 …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 {#1265 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1266 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1267 …2} "content" => Illuminate\Database\Eloquent\Collection {#1270 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1276 …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 {#1269 #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" => 1113972 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/22-12" "field_display_title" => "22.12" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1268 …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 {#1275 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1277 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1273 …2} "content" => Illuminate\Database\Eloquent\Collection {#1272 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1290 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113972 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/22-12" "field_display_title" => "22.12" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1268 …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 {#1275 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1277 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1273 …2} "content" => Illuminate\Database\Eloquent\Collection {#1272 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1290 …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 => "*" ] } 6 => App\Models\Task {#1274 #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" => 1113973 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/22-13" "field_display_title" => "22.13" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1271 …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 {#1289 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1291 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1287 …2} "content" => Illuminate\Database\Eloquent\Collection {#1286 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1304 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113973 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/22-13" "field_display_title" => "22.13" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1271 …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 {#1289 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1291 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1287 …2} "content" => Illuminate\Database\Eloquent\Collection {#1286 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1304 …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 => "*" ] } 7 => App\Models\Task {#1288 #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" => 1113974 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/22-14" "field_display_title" => "22.14" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1285 …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 {#1303 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1305 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1301 …2} "content" => Illuminate\Database\Eloquent\Collection {#1300 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1318 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113974 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/22-14" "field_display_title" => "22.14" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1285 …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 {#1303 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1305 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1301 …2} "content" => Illuminate\Database\Eloquent\Collection {#1300 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1318 …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 => "*" ] } 8 => App\Models\Task {#1302 #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" => 1113975 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/22-15" "field_display_title" => "22.15" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1299 …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 {#1317 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1319 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1315 …2} "content" => Illuminate\Database\Eloquent\Collection {#1314 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1332 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113975 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/22-15" "field_display_title" => "22.15" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1299 …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 {#1317 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1319 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1315 …2} "content" => Illuminate\Database\Eloquent\Collection {#1314 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1332 …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 => "*" ] } 9 => App\Models\Task {#1316 #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" => 1113976 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/22-16" "field_display_title" => "22.16" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1313 …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 {#1331 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1333 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1329 …2} "content" => Illuminate\Database\Eloquent\Collection {#1328 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1346 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113976 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/22-16" "field_display_title" => "22.16" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1313 …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 {#1331 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1333 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1329 …2} "content" => Illuminate\Database\Eloquent\Collection {#1328 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1346 …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 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1098458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/page-121" "field_display_title" => "121" "field_folder" => "1" "field_image_name" => "121" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1040} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1103} "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 {#1112} "content" => Illuminate\Database\Eloquent\Collection {#1128} "next" => array:2 [ "refs" => "1098459" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1098457" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1198} ] #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" => 1113970 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/22-10" "field_display_title" => "22.10" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037} 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№22.10 (с. 121)
Условие. №22.10 (с. 121)
скриншот условия
22.10. a) $f(x) = \frac{1}{x} + x^2$, $[0,5; 1];$
б) $f(x) = \sqrt{2 - x - x^2}$, $[-1; 0].$
Решение. №22.10 (с. 121)
Решение 2. №22.10 (с. 121)
а) Для нахождения наибольшего и наименьшего значений функции $f(x) = \frac{1}{x} + x^2$ на отрезке $[0.5; 1]$ необходимо найти значения функции на концах отрезка и в критических точках, принадлежащих этому отрезку, а затем сравнить их.
1. Найдём производную функции $f(x)$:
$f'(x) = \left(\frac{1}{x} + x^2\right)' = (x^{-1} + x^2)' = -1 \cdot x^{-2} + 2x = -\frac{1}{x^2} + 2x$.
2. Найдём критические точки, приравняв производную к нулю:
$-\frac{1}{x^2} + 2x = 0$
$2x = \frac{1}{x^2}$
$2x^3 = 1$
$x^3 = \frac{1}{2}$
$x = \sqrt[3]{\frac{1}{2}} = \frac{1}{\sqrt[3]{2}}$.
3. Проверим, принадлежит ли найденная критическая точка $x = \frac{1}{\sqrt[3]{2}}$ отрезку $[0.5; 1]$.
Поскольку $1^3 = 1$ и $2^3 = 8$, то $1 < \sqrt[3]{2} < 2$. Разделив все части на $\sqrt[3]{2}$ и на 2, получим $\frac{1}{2} < \frac{1}{\sqrt[3]{2}} < 1$.
Следовательно, $0.5 < \frac{1}{\sqrt[3]{2}} < 1$, и критическая точка принадлежит заданному отрезку.
4. Вычислим значения функции в критической точке и на концах отрезка:
На левом конце отрезка: $f(0.5) = \frac{1}{0.5} + (0.5)^2 = 2 + 0.25 = 2.25$.
На правом конце отрезка: $f(1) = \frac{1}{1} + 1^2 = 1 + 1 = 2$.
В критической точке: $f\left(\frac{1}{\sqrt[3]{2}}\right) = \frac{1}{1/\sqrt[3]{2}} + \left(\frac{1}{\sqrt[3]{2}}\right)^2 = \sqrt[3]{2} + \frac{1}{\sqrt[3]{4}} = \frac{\sqrt[3]{2}\sqrt[3]{4}+1}{\sqrt[3]{4}} = \frac{\sqrt[3]{8}+1}{\sqrt[3]{4}} = \frac{3}{\sqrt[3]{4}} = \frac{3\sqrt[3]{2}}{2}$.
5. Сравним полученные значения: $2.25$, $2$ и $\frac{3\sqrt[3]{2}}{2}$.
Чтобы сравнить $\frac{3\sqrt[3]{2}}{2}$ и $2$, сравним $3\sqrt[3]{2}$ и $4$. Возведём обе части в куб: $(3\sqrt[3]{2})^3 = 27 \cdot 2 = 54$ и $4^3 = 64$. Так как $54 < 64$, то $3\sqrt[3]{2} < 4$, и $\frac{3\sqrt[3]{2}}{2} < 2$.
Таким образом, из трёх значений $2.25$, $2$ и $\frac{3\sqrt[3]{2}}{2}$ наибольшее — это $2.25$, а наименьшее — $\frac{3\sqrt[3]{2}}{2}$.
Наибольшее значение функции: $\max_{x \in [0.5; 1]} f(x) = f(0.5) = 2.25$.
Наименьшее значение функции: $\min_{x \in [0.5; 1]} f(x) = f\left(\frac{1}{\sqrt[3]{2}}\right) = \frac{3\sqrt[3]{2}}{2}$.
Ответ: наименьшее значение функции равно $\frac{3\sqrt[3]{2}}{2}$, наибольшее значение функции равно $2.25$.
б) Для нахождения наибольшего и наименьшего значений функции $f(x) = \sqrt{2 - x - x^2}$ на отрезке $[-1; 0]$ будем использовать тот же алгоритм.
1. Сначала найдём область определения функции. Выражение под корнем должно быть неотрицательным: $2 - x - x^2 \ge 0$, или $x^2 + x - 2 \le 0$. Корнями уравнения $x^2 + x - 2 = 0$ являются $x_1 = -2$ и $x_2 = 1$. Неравенство выполняется на отрезке $[-2; 1]$. Заданный отрезок $[-1; 0]$ полностью входит в область определения, поэтому функция на нём непрерывна.
2. Найдём производную функции $f(x)$:
$f'(x) = \left(\sqrt{2 - x - x^2}\right)' = \frac{1}{2\sqrt{2 - x - x^2}} \cdot (2 - x - x^2)' = \frac{-1 - 2x}{2\sqrt{2 - x - x^2}}$.
3. Найдём критические точки из условия $f'(x) = 0$:
$\frac{-1 - 2x}{2\sqrt{2 - x - x^2}} = 0$
Для этого числитель должен быть равен нулю: $-1 - 2x = 0$, откуда $x = -0.5$.
Эта точка принадлежит отрезку $[-1; 0]$.
4. Вычислим значения функции в критической точке и на концах отрезка:
$f(-1) = \sqrt{2 - (-1) - (-1)^2} = \sqrt{2 + 1 - 1} = \sqrt{2}$.
$f(0) = \sqrt{2 - 0 - 0^2} = \sqrt{2}$.
$f(-0.5) = \sqrt{2 - (-0.5) - (-0.5)^2} = \sqrt{2 + 0.5 - 0.25} = \sqrt{2.25} = 1.5$.
5. Сравним полученные значения: $\sqrt{2}$ и $1.5$.
Так как $1.5 = \sqrt{2.25}$, а $2.25 > 2$, то $1.5 > \sqrt{2}$.
Наибольшее значение функции: $\max_{x \in [-1; 0]} f(x) = f(-0.5) = 1.5$.
Наименьшее значение функции: $\min_{x \in [-1; 0]} f(x) = f(-1) = f(0) = \sqrt{2}$.
Ответ: наименьшее значение функции равно $\sqrt{2}$, наибольшее значение функции равно $1.5$.
Другие задания:
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