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Номер 19.9, страница 109 - 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 19. Признаки возрастания и убывания функции - номер 19.9, страница 109.
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" => 1113933 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => "110" "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/19-9" "field_display_title" => "19.9" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#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" "field_prepositional_case" => "учебнике" ] "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 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "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 [ "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" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "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" => "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 [ "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" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "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" => "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" => 1113581 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Признаки возрастания и убывания функции" "field_branch_order" => "19" "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" => "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 {#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" => 1113581 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Признаки возрастания и убывания функции" "field_branch_order" => "19" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1121} "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 {#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" => 1290874 "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" => "1113933" "type" => "task" ] "text" => "<p><strong>19.9.</strong> Докажите, что на множестве действительных чисел функция $f(x)$ является убывающей, а $g(x)$ — возрастающей:</p><p><strong>а)</strong> $f(x) = 5 - x^7;$</p><p><strong>б)</strong> $g(x) = 4 + \frac{2}{3} x^3;$</p><p><strong>в)</strong> $f(x) = -8x - \sin2x;$</p><p><strong>г)</strong> $g(x) = -\cos6x + 7x.$</p>" "img" => array:2 [ 0 => array:5 [ "name" => "9-1.jpg" "alt" => null "width" => "2751" "height" => 314 "path" => "/media/algebra_10/Abylkasymova-u/0-19/9-1.webp?ts=1753263114" ] 1 => array:5 [ "name" => "9-2.jpg" "alt" => null "width" => "2436" "height" => 372 "path" => "/media/algebra_10/Abylkasymova-u/0-19/9-2.webp?ts=1753263114" ] ] ] #original: array:7 [ "id" => 1290874 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1087} "task" => array:2 [ "refs" => "1113933" "type" => "task" ] "text" => "<p><strong>19.9.</strong> Докажите, что на множестве действительных чисел функция $f(x)$ является убывающей, а $g(x)$ — возрастающей:</p><p><strong>а)</strong> $f(x) = 5 - x^7;$</p><p><strong>б)</strong> $g(x) = 4 + \frac{2}{3} x^3;$</p><p><strong>в)</strong> $f(x) = -8x - \sin2x;$</p><p><strong>г)</strong> $g(x) = -\cos6x + 7x.$</p>" "img" => array:2 [ 0 => array:5 [ "name" => "9-1.jpg" "alt" => null "width" => "2751" "height" => 314 "path" => "/media/algebra_10/Abylkasymova-u/0-19/9-1.webp?ts=1753263114" ] 1 => array:5 [ "name" => "9-2.jpg" "alt" => null "width" => "2436" "height" => 372 "path" => "/media/algebra_10/Abylkasymova-u/0-19/9-2.webp?ts=1753263114" ] ] ] #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" => 1291388 "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" => "1113933" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "9-1.jpg" "alt" => null "width" => "1848" "height" => 1675 "path" => "/media/algebra_10/Abylkasymova-u/1-19/9-1.webp?ts=1753263319" ] 1 => array:5 [ "name" => "9-2.jpg" "alt" => null "width" => "1558" "height" => 582 "path" => "/media/algebra_10/Abylkasymova-u/1-19/9-2.webp?ts=1753263319" ] ] ] #original: array:6 [ "id" => 1291388 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1079} "task" => array:2 [ "refs" => "1113933" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "9-1.jpg" "alt" => null "width" => "1848" "height" => 1675 "path" => "/media/algebra_10/Abylkasymova-u/1-19/9-1.webp?ts=1753263319" ] 1 => array:5 [ "name" => "9-2.jpg" "alt" => null "width" => "1558" "height" => 582 "path" => "/media/algebra_10/Abylkasymova-u/1-19/9-2.webp?ts=1753263319" ] ] ] #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" => 1549517 "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" => "1113933" "type" => "task" ] "text" => "<p><strong>а)</strong> Чтобы доказать, что функция является убывающей на множестве действительных чисел, необходимо найти ее производную и показать, что она неположительна ($f'(x) \le 0$) для всех действительных $x$, при этом обращаясь в ноль лишь в отдельных точках. Дана функция $f(x) = 5 - x^7$. Найдем ее производную: $f'(x) = (5 - x^7)' = 0 - 7x^{7-1} = -7x^6$. Выражение $x^6$ является четной степенью $x$, поэтому оно всегда неотрицательно, то есть $x^6 \ge 0$ для любого действительного числа $x$. Следовательно, производная $f'(x) = -7x^6$ всегда неположительна: $f'(x) \le 0$. Равенство $f'(x) = 0$ выполняется только в одной точке, при $x=0$. Так как производная функции неположительна на всей числовой оси и обращается в ноль лишь в изолированной точке, функция $f(x)$ является строго убывающей на множестве действительных чисел.</p><p><strong>Ответ:</strong> Доказано, что функция $f(x) = 5 - x^7$ является убывающей на множестве действительных чисел.</p><p><strong>б)</strong> Чтобы доказать, что функция является возрастающей на множестве действительных чисел, необходимо найти ее производную и показать, что она неотрицательна ($g'(x) \ge 0$) для всех действительных $x$, при этом обращаясь в ноль лишь в отдельных точках. Дана функция $g(x) = 4 + \frac{2}{3}x^3$. Найдем ее производную: $g'(x) = (4 + \frac{2}{3}x^3)' = 0 + \frac{2}{3} \cdot 3x^{3-1} = 2x^2$. Выражение $x^2$ всегда неотрицательно: $x^2 \ge 0$ для любого действительного числа $x$. Следовательно, производная $g'(x) = 2x^2$ всегда неотрицательна: $g'(x) \ge 0$. Равенство $g'(x) = 0$ выполняется только в одной точке, при $x=0$. Так как производная функции неотрицательна на всей числовой оси и обращается в ноль лишь в изолированной точке, функция $g(x)$ является строго возрастающей на множестве действительных чисел.</p><p><strong>Ответ:</strong> Доказано, что функция $g(x) = 4 + \frac{2}{3}x^3$ является возрастающей на множестве действительных чисел.</p><p><strong>в)</strong> Чтобы доказать, что функция $f(x) = -8x - \sin(2x)$ является убывающей, найдем ее производную и исследуем ее знак. $f'(x) = (-8x - \sin(2x))' = -8 - (\sin(2x))' = -8 - \cos(2x) \cdot (2x)' = -8 - 2\cos(2x)$. Область значений функции косинус есть отрезок $[-1, 1]$, то есть $-1 \le \cos(2x) \le 1$ для любого $x \in \mathbb{R}$. Умножим все части неравенства на 2: $-2 \le 2\cos(2x) \le 2$. Теперь оценим выражение для производной $f'(x) = -8 - 2\cos(2x)$. Максимальное значение производной достигается, когда $2\cos(2x)$ минимально, то есть равно -2: $f'_{\text{max}} = -8 - (-2) = -6$. Минимальное значение производной достигается, когда $2\cos(2x)$ максимально, то есть равно 2: $f'_{\text{min}} = -8 - 2 = -10$. Таким образом, для любого действительного $x$ выполняется неравенство $-10 \le f'(x) \le -6$. Это означает, что производная $f'(x)$ всегда отрицательна ($f'(x) < 0$). Поскольку производная функции строго отрицательна на всей числовой оси, функция $f(x)$ является убывающей на множестве действительных чисел.</p><p><strong>Ответ:</strong> Доказано, что функция $f(x) = -8x - \sin(2x)$ является убывающей на множестве действительных чисел.</p><p><strong>г)</strong> Чтобы доказать, что функция $g(x) = -\cos(6x) + 7x$ является возрастающей, найдем ее производную и исследуем ее знак. $g'(x) = (-\cos(6x) + 7x)' = -(-\sin(6x)) \cdot (6x)' + 7 = 6\sin(6x) + 7$. Область значений функции синус есть отрезок $[-1, 1]$, то есть $-1 \le \sin(6x) \le 1$ для любого $x \in \mathbb{R}$. Умножим все части неравенства на 6: $-6 \le 6\sin(6x) \le 6$. Прибавим 7 ко всем частям неравенства, чтобы оценить $g'(x) = 6\sin(6x) + 7$: $-6 + 7 \le 6\sin(6x) + 7 \le 6 + 7$. $1 \le g'(x) \le 13$. Это означает, что производная $g'(x)$ всегда положительна ($g'(x) > 0$) для любого действительного числа $x$. Поскольку производная функции строго положительна на всей числовой оси, функция $g(x)$ является возрастающей на множестве действительных чисел.</p><p><strong>Ответ:</strong> Доказано, что функция $g(x) = -\cos(6x) + 7x$ является возрастающей на множестве действительных чисел.</p>" ] #original: array:6 [ "id" => 1549517 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108} "task" => array:2 [ "refs" => "1113933" "type" => "task" ] "text" => "<p><strong>а)</strong> Чтобы доказать, что функция является убывающей на множестве действительных чисел, необходимо найти ее производную и показать, что она неположительна ($f'(x) \le 0$) для всех действительных $x$, при этом обращаясь в ноль лишь в отдельных точках. Дана функция $f(x) = 5 - x^7$. Найдем ее производную: $f'(x) = (5 - x^7)' = 0 - 7x^{7-1} = -7x^6$. Выражение $x^6$ является четной степенью $x$, поэтому оно всегда неотрицательно, то есть $x^6 \ge 0$ для любого действительного числа $x$. Следовательно, производная $f'(x) = -7x^6$ всегда неположительна: $f'(x) \le 0$. Равенство $f'(x) = 0$ выполняется только в одной точке, при $x=0$. Так как производная функции неположительна на всей числовой оси и обращается в ноль лишь в изолированной точке, функция $f(x)$ является строго убывающей на множестве действительных чисел.</p><p><strong>Ответ:</strong> Доказано, что функция $f(x) = 5 - x^7$ является убывающей на множестве действительных чисел.</p><p><strong>б)</strong> Чтобы доказать, что функция является возрастающей на множестве действительных чисел, необходимо найти ее производную и показать, что она неотрицательна ($g'(x) \ge 0$) для всех действительных $x$, при этом обращаясь в ноль лишь в отдельных точках. Дана функция $g(x) = 4 + \frac{2}{3}x^3$. Найдем ее производную: $g'(x) = (4 + \frac{2}{3}x^3)' = 0 + \frac{2}{3} \cdot 3x^{3-1} = 2x^2$. Выражение $x^2$ всегда неотрицательно: $x^2 \ge 0$ для любого действительного числа $x$. Следовательно, производная $g'(x) = 2x^2$ всегда неотрицательна: $g'(x) \ge 0$. Равенство $g'(x) = 0$ выполняется только в одной точке, при $x=0$. Так как производная функции неотрицательна на всей числовой оси и обращается в ноль лишь в изолированной точке, функция $g(x)$ является строго возрастающей на множестве действительных чисел.</p><p><strong>Ответ:</strong> Доказано, что функция $g(x) = 4 + \frac{2}{3}x^3$ является возрастающей на множестве действительных чисел.</p><p><strong>в)</strong> Чтобы доказать, что функция $f(x) = -8x - \sin(2x)$ является убывающей, найдем ее производную и исследуем ее знак. $f'(x) = (-8x - \sin(2x))' = -8 - (\sin(2x))' = -8 - \cos(2x) \cdot (2x)' = -8 - 2\cos(2x)$. Область значений функции косинус есть отрезок $[-1, 1]$, то есть $-1 \le \cos(2x) \le 1$ для любого $x \in \mathbb{R}$. Умножим все части неравенства на 2: $-2 \le 2\cos(2x) \le 2$. Теперь оценим выражение для производной $f'(x) = -8 - 2\cos(2x)$. Максимальное значение производной достигается, когда $2\cos(2x)$ минимально, то есть равно -2: $f'_{\text{max}} = -8 - (-2) = -6$. Минимальное значение производной достигается, когда $2\cos(2x)$ максимально, то есть равно 2: $f'_{\text{min}} = -8 - 2 = -10$. Таким образом, для любого действительного $x$ выполняется неравенство $-10 \le f'(x) \le -6$. Это означает, что производная $f'(x)$ всегда отрицательна ($f'(x) < 0$). Поскольку производная функции строго отрицательна на всей числовой оси, функция $f(x)$ является убывающей на множестве действительных чисел.</p><p><strong>Ответ:</strong> Доказано, что функция $f(x) = -8x - \sin(2x)$ является убывающей на множестве действительных чисел.</p><p><strong>г)</strong> Чтобы доказать, что функция $g(x) = -\cos(6x) + 7x$ является возрастающей, найдем ее производную и исследуем ее знак. $g'(x) = (-\cos(6x) + 7x)' = -(-\sin(6x)) \cdot (6x)' + 7 = 6\sin(6x) + 7$. Область значений функции синус есть отрезок $[-1, 1]$, то есть $-1 \le \sin(6x) \le 1$ для любого $x \in \mathbb{R}$. Умножим все части неравенства на 6: $-6 \le 6\sin(6x) \le 6$. Прибавим 7 ко всем частям неравенства, чтобы оценить $g'(x) = 6\sin(6x) + 7$: $-6 + 7 \le 6\sin(6x) + 7 \le 6 + 7$. $1 \le g'(x) \le 13$. Это означает, что производная $g'(x)$ всегда положительна ($g'(x) > 0$) для любого действительного числа $x$. Поскольку производная функции строго положительна на всей числовой оси, функция $g(x)$ является возрастающей на множестве действительных чисел.</p><p><strong>Ответ:</strong> Доказано, что функция $g(x) = -\cos(6x) + 7x$ является возрастающей на множестве действительных чисел.</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" => "1113934" "type" => "task" ] "previous" => array:2 [ "refs" => "1113932" "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" => 1098446 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/page-109" "field_display_title" => "109" "field_folder" => "1" "field_image_name" => "109" "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" => 1261871 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1129 #items: array:1 [ …1] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1098446" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #original: array:6 [ "id" => 1261871 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1129} "book_page" => array:2 [ "refs" => "1098446" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #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" => "1098447" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1098445" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1197 #items: array:9 [ 0 => App\Models\Task {#1213 #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" => 1113925 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/19-1" "field_display_title" => "19.1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1214 #items: array:1 [ …1] #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 {#1215 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1216 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1217 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1220 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113926" "type" => "task" ] "previous" => array:2 [ "refs" => "1113924" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1226 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098446" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113925 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/19-1" "field_display_title" => "19.1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1214} "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 {#1215} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1216} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1217} "content" => Illuminate\Database\Eloquent\Collection {#1220} "next" => array:2 [ "refs" => "1113926" "type" => "task" ] "previous" => array:2 [ "refs" => "1113924" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1226} "page" => array:2 [ "refs" => "1098446" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1219 #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" => 1113926 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/19-2" "field_display_title" => "19.2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1218 #items: array:1 [ …1] #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 {#1225 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1227 #items: array:1 [ …1] …1 } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1223 …2} "content" => Illuminate\Database\Eloquent\Collection {#1222 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1240 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113926 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/19-2" "field_display_title" => "19.2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1218} "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 {#1225} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1227 …1} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1223 …2} "content" => Illuminate\Database\Eloquent\Collection {#1222 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1240 …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 {#1224 #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" => 1113927 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/19-3" "field_display_title" => "19.3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1221 …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 {#1239 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1241 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1237 …2} "content" => Illuminate\Database\Eloquent\Collection {#1236 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1254 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113927 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/19-3" "field_display_title" => "19.3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1221 …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 {#1239 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1241 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1237 …2} "content" => Illuminate\Database\Eloquent\Collection {#1236 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1254 …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 {#1238 #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" => 1113928 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/19-4" "field_display_title" => "19.4" "field_outside_task" => null "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 {#1253 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1255 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1251 …2} "content" => Illuminate\Database\Eloquent\Collection {#1250 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1268 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113928 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/19-4" "field_display_title" => "19.4" "field_outside_task" => null "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 {#1253 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1255 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1251 …2} "content" => Illuminate\Database\Eloquent\Collection {#1250 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1268 …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 {#1252 #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" => 1113929 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/19-5" "field_display_title" => "19.5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1249 …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 {#1267 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1269 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1265 …2} "content" => Illuminate\Database\Eloquent\Collection {#1264 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1282 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113929 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/19-5" "field_display_title" => "19.5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1249 …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 {#1267 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1269 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1265 …2} "content" => Illuminate\Database\Eloquent\Collection {#1264 …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 => "*" ] } 5 => App\Models\Task {#1266 #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" => 1113930 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/19-6" "field_display_title" => "19.6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1263 …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 {#1281 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1283 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1279 …2} "content" => Illuminate\Database\Eloquent\Collection {#1278 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1296 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113930 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/19-6" "field_display_title" => "19.6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1263 …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 {#1281 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1283 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1279 …2} "content" => Illuminate\Database\Eloquent\Collection {#1278 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1296 …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 {#1280 #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" => 1113931 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/19-7" "field_display_title" => "19.7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1277 …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 {#1295 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1297 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1293 …2} "content" => Illuminate\Database\Eloquent\Collection {#1292 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1310 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113931 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/19-7" "field_display_title" => "19.7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1277 …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 {#1295 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1297 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1293 …2} "content" => Illuminate\Database\Eloquent\Collection {#1292 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1310 …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 {#1294 #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" => 1113932 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/19-8" "field_display_title" => "19.8" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1291 …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 {#1309 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1311 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1307 …2} "content" => Illuminate\Database\Eloquent\Collection {#1306 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1324 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113932 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/19-8" "field_display_title" => "19.8" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1291 …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 {#1309 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1311 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1307 …2} "content" => Illuminate\Database\Eloquent\Collection {#1306 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1324 …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 {#1308 #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" => 1113933 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => "110" "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/19-9" "field_display_title" => "19.9" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1305 …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 {#1323 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1325 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1321 …2} "content" => Illuminate\Database\Eloquent\Collection {#1319 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1322 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113933 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => "110" "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/19-9" "field_display_title" => "19.9" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1305 …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 {#1323 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1325 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1321 …2} "content" => Illuminate\Database\Eloquent\Collection {#1319 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1322 …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" => 1098446 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/page-109" "field_display_title" => "109" "field_folder" => "1" "field_image_name" => "109" "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" => "1098447" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1098445" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1197} ] #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" => 1113933 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "109" "field_page_end" => "110" "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/19-9" "field_display_title" => "19.9" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1046} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1105} "content" => Illuminate\Database\Eloquent\Collection {#1102} "next" => array:2 [ "refs" => "1113934" "type" => "task" ] "previous" => array:2 [ "refs" => "1113932" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1098} "page" => array:2 [ "refs" => "1098446" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] 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№19.9 (с. 109)
Условие. №19.9 (с. 109)
скриншот условия
19.9. Докажите, что на множестве действительных чисел функция $f(x)$ является убывающей, а $g(x)$ — возрастающей:
а) $f(x) = 5 - x^7;$
б) $g(x) = 4 + \frac{2}{3} x^3;$
в) $f(x) = -8x - \sin2x;$
г) $g(x) = -\cos6x + 7x.$
Решение. №19.9 (с. 109)
Решение 2. №19.9 (с. 109)
а) Чтобы доказать, что функция является убывающей на множестве действительных чисел, необходимо найти ее производную и показать, что она неположительна ($f'(x) \le 0$) для всех действительных $x$, при этом обращаясь в ноль лишь в отдельных точках. Дана функция $f(x) = 5 - x^7$. Найдем ее производную: $f'(x) = (5 - x^7)' = 0 - 7x^{7-1} = -7x^6$. Выражение $x^6$ является четной степенью $x$, поэтому оно всегда неотрицательно, то есть $x^6 \ge 0$ для любого действительного числа $x$. Следовательно, производная $f'(x) = -7x^6$ всегда неположительна: $f'(x) \le 0$. Равенство $f'(x) = 0$ выполняется только в одной точке, при $x=0$. Так как производная функции неположительна на всей числовой оси и обращается в ноль лишь в изолированной точке, функция $f(x)$ является строго убывающей на множестве действительных чисел.
Ответ: Доказано, что функция $f(x) = 5 - x^7$ является убывающей на множестве действительных чисел.
б) Чтобы доказать, что функция является возрастающей на множестве действительных чисел, необходимо найти ее производную и показать, что она неотрицательна ($g'(x) \ge 0$) для всех действительных $x$, при этом обращаясь в ноль лишь в отдельных точках. Дана функция $g(x) = 4 + \frac{2}{3}x^3$. Найдем ее производную: $g'(x) = (4 + \frac{2}{3}x^3)' = 0 + \frac{2}{3} \cdot 3x^{3-1} = 2x^2$. Выражение $x^2$ всегда неотрицательно: $x^2 \ge 0$ для любого действительного числа $x$. Следовательно, производная $g'(x) = 2x^2$ всегда неотрицательна: $g'(x) \ge 0$. Равенство $g'(x) = 0$ выполняется только в одной точке, при $x=0$. Так как производная функции неотрицательна на всей числовой оси и обращается в ноль лишь в изолированной точке, функция $g(x)$ является строго возрастающей на множестве действительных чисел.
Ответ: Доказано, что функция $g(x) = 4 + \frac{2}{3}x^3$ является возрастающей на множестве действительных чисел.
в) Чтобы доказать, что функция $f(x) = -8x - \sin(2x)$ является убывающей, найдем ее производную и исследуем ее знак. $f'(x) = (-8x - \sin(2x))' = -8 - (\sin(2x))' = -8 - \cos(2x) \cdot (2x)' = -8 - 2\cos(2x)$. Область значений функции косинус есть отрезок $[-1, 1]$, то есть $-1 \le \cos(2x) \le 1$ для любого $x \in \mathbb{R}$. Умножим все части неравенства на 2: $-2 \le 2\cos(2x) \le 2$. Теперь оценим выражение для производной $f'(x) = -8 - 2\cos(2x)$. Максимальное значение производной достигается, когда $2\cos(2x)$ минимально, то есть равно -2: $f'_{\text{max}} = -8 - (-2) = -6$. Минимальное значение производной достигается, когда $2\cos(2x)$ максимально, то есть равно 2: $f'_{\text{min}} = -8 - 2 = -10$. Таким образом, для любого действительного $x$ выполняется неравенство $-10 \le f'(x) \le -6$. Это означает, что производная $f'(x)$ всегда отрицательна ($f'(x) < 0$). Поскольку производная функции строго отрицательна на всей числовой оси, функция $f(x)$ является убывающей на множестве действительных чисел.
Ответ: Доказано, что функция $f(x) = -8x - \sin(2x)$ является убывающей на множестве действительных чисел.
г) Чтобы доказать, что функция $g(x) = -\cos(6x) + 7x$ является возрастающей, найдем ее производную и исследуем ее знак. $g'(x) = (-\cos(6x) + 7x)' = -(-\sin(6x)) \cdot (6x)' + 7 = 6\sin(6x) + 7$. Область значений функции синус есть отрезок $[-1, 1]$, то есть $-1 \le \sin(6x) \le 1$ для любого $x \in \mathbb{R}$. Умножим все части неравенства на 6: $-6 \le 6\sin(6x) \le 6$. Прибавим 7 ко всем частям неравенства, чтобы оценить $g'(x) = 6\sin(6x) + 7$: $-6 + 7 \le 6\sin(6x) + 7 \le 6 + 7$. $1 \le g'(x) \le 13$. Это означает, что производная $g'(x)$ всегда положительна ($g'(x) > 0$) для любого действительного числа $x$. Поскольку производная функции строго положительна на всей числовой оси, функция $g(x)$ является возрастающей на множестве действительных чисел.
Ответ: Доказано, что функция $g(x) = -\cos(6x) + 7x$ является возрастающей на множестве действительных чисел.
Другие задания:
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