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Номер 8, страница 127, часть 2 - гдз по алгебре 10 класс учебник Пак, Ардакулы
Авторы: Пак О. В., Ардакулы Д., Ескендирова Е. В.
Тип: Учебник
Издательство: Алматыкітап баспасы
Год издания: 2019 - 2026
Часть: 2
ISBN: 978-601-01-3958-9
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 10 классе
Часть 2. Глава 6. Применение производной. Параграф 5. Исследование функций и построение графиков. 5.2. Исследование функций и построение графиков - номер 8, страница 127.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1120431 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "127" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-5-2zad-8" "field_display_title" => "8" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1072 #items: array:1 [ 0 => App\Models\Branch {#1034 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:1 [ 0 => App\Models\Term {#1038 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #original: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1040 #items: array:1 [ 0 => App\Models\Book {#1041 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4295 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1042 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4295 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1042 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1040} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1069} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1144 #items: array:4 [ 0 => App\Models\Branch {#1154 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1158 #items: array:1 [ 0 => App\Models\Term {#1155 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #original: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1187 #items: array:1 [ 0 => App\Models\Book {#1157 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4295 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1159 #items: array:1 [ 0 => App\Models\Term {#1156 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1160 #items: array:1 [ 0 => App\Models\Term {#1161 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1162 #items: array:1 [ 0 => App\Models\Term {#1163 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 7006 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматыкітап баспасы" "field_cases" => array:6 [ …6] "field_translit" => "almatykitap baspasy" ] #original: array:6 [ "id" => 7006 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматыкітап баспасы" "field_cases" => array:6 [ …6] "field_translit" => "almatykitap baspasy" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1164 #items: array:3 [ 0 => App\Models\Term {#1165 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Пак" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Олег" "field_patronymic" => "Владимирович" "field_surname_rp" => null ] #original: array:12 [ "id" => 7007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Пак" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Олег" "field_patronymic" => "Владимирович" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1166 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7008 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ардакулы" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Дархан" "field_patronymic" => null "field_surname_rp" => null ] #original: array:12 [ "id" => 7008 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ардакулы" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Дархан" "field_patronymic" => null "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Term {#1167 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7009 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ескендирова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Елена" "field_patronymic" => "Викторовна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7009 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ескендирова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Елена" "field_patronymic" => "Викторовна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1168 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1169 #items: array:1 [ 0 => App\Models\Term {#1170 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1171 #items: array:1 [ 0 => App\Models\Term {#1172 #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 {#1173 #items: array:1 [ 0 => App\Models\Term {#1174 #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 {#1175 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1176 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1177 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1178 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1179 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1180 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1181 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1182 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1183 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1184 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #original: array:50 [ "id" => 4295 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1159} "field_class" => Illuminate\Database\Eloquent\Collection {#1160} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1162} "field_author" => Illuminate\Database\Eloquent\Collection {#1164} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1168} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1169} "field_country" => Illuminate\Database\Eloquent\Collection {#1171} "field_city" => Illuminate\Database\Eloquent\Collection {#1173} "field_series" => Illuminate\Database\Eloquent\Collection {#1175} "field_umk" => Illuminate\Database\Eloquent\Collection {#1176} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1177} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1178} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1179} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1180} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1181} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1182} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1183} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1184} "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 {#1185 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1158} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1187} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1185} ] #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 {#1186 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119442 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Применение производной" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1191 #items: array:1 [ 0 => App\Models\Term {#1188 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "76" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1190 #items: array:1 [ 0 => App\Models\Book {#1157} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1189 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119442 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Применение производной" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1191} "field_page_start" => "76" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1190} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1189} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Branch {#1192 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119453 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Исследование функций и построение графиков" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1196 #items: array:1 [ 0 => App\Models\Term {#1193 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "114" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1195 #items: array:1 [ 0 => App\Models\Book {#1157} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1236 #items: array:1 [ 0 => App\Models\Branch {#1194 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119442 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Применение производной" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1198 …2} "field_page_start" => "76" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1199 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1228 …2} ] #original: array:24 [ "id" => 1119442 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Применение производной" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1198 …2} "field_page_start" => "76" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1199 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1228 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119453 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => 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[] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Branch {#1234 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119454 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "5.2. 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(4)</strong> Исследуйте функцию $f(x) = 4\cos\frac{x}{2} - 2\cos x$ и постройте ее график.</p><p><strong>9.</strong> (2) Дана функция $g(x) = 2\sin x$ определенная на промежутке</p>" "img" => array:1 [ 0 => array:5 [ "name" => "8-1.jpg" "alt" => null "width" => "1637" "height" => 196 "path" => "/media/algebra_10/baspasy-u19/0-6-5-2zad/8-1.webp?ts=1753534949" ] ] ] #original: array:7 [ "id" => 1301116 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1122} "task" => array:2 [ "refs" => "1120431" "type" => "task" ] "text" => "<p><strong>8. (4)</strong> Исследуйте функцию $f(x) = 4\cos\frac{x}{2} - 2\cos x$ и постройте ее график.</p><p><strong>9.</strong> (2) Дана функция $g(x) = 2\sin x$ определенная на промежутке</p>" "img" => array:1 [ 0 => array:5 [ "name" => "8-1.jpg" "alt" => null "width" => "1637" "height" => 196 "path" => "/media/algebra_10/baspasy-u19/0-6-5-2zad/8-1.webp?ts=1753534949" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1124 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1552152 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\Edition {#1125 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5659 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1120 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1115 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5659 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1120 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1115 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1120431" "type" => "task" ] "text" => "<p>Проведем полное исследование функции $f(x) = 4\cos\frac{x}{2} - 2\cos x$ и построим ее график, выполнив следующие шаги.</p><p><strong>1. Область определения</strong></p><p>Функция состоит из комбинации тригонометрических функций $\cos\frac{x}{2}$ и $\cos x$, которые определены для всех действительных чисел. Следовательно, область определения функции $f(x)$ — это множество всех действительных чисел.</p><p><strong>Ответ:</strong> $D(f) = (-\infty; +\infty)$.</p><p><strong>2. Четность и периодичность</strong></p><p>Проверим функцию на четность: $f(-x) = 4\cos\left(-\frac{x}{2}\right) - 2\cos(-x) = 4\cos\frac{x}{2} - 2\cos x = f(x)$, так как функция косинуса является четной. Таким образом, функция $f(x)$ — четная, ее график симметричен относительно оси ординат ($Oy$).</p><p>Период для слагаемого $4\cos\frac{x}{2}$ равен $T_1 = \frac{2\pi}{1/2} = 4\pi$. Период для слагаемого $-2\cos x$ равен $T_2 = 2\pi$. Наименьший общий период функции $f(x)$ равен наименьшему общему кратному периодов $T_1$ и $T_2$, то есть $T = \text{НОК}(4\pi, 2\pi) = 4\pi$.</p><p><strong>Ответ:</strong> Функция является четной и периодической с основным периодом $T = 4\pi$.</p><p><strong>3. Точки пересечения с осями координат</strong></p><p>С осью $Oy$: при $x=0$ имеем $f(0) = 4\cos(0) - 2\cos(0) = 4 - 2 = 2$. Точка пересечения — $(0; 2)$.</p><p>С осью $Ox$: решаем уравнение $f(x) = 0$, то есть $4\cos\frac{x}{2} - 2\cos x = 0$. Используем формулу косинуса двойного угла $\cos x = 2\cos^2\frac{x}{2} - 1$:</p><p>$4\cos\frac{x}{2} - 2(2\cos^2\frac{x}{2} - 1) = 0$, что приводит к квадратному уравнению относительно $\cos\frac{x}{2}$: $2\cos^2\frac{x}{2} - 2\cos\frac{x}{2} - 1 = 0$.</p><p>Решая его, получаем $\cos\frac{x}{2} = \frac{1 \pm \sqrt{3}}{2}$. Так как $|\cos \alpha| \le 1$, подходит только корень $\cos\frac{x}{2} = \frac{1 - \sqrt{3}}{2}$.</p><p>Отсюда $x = \pm 2\arccos\left(\frac{1 - \sqrt{3}}{2}\right) + 4\pi n$, где $n \in \mathbb{Z}$.</p><p><strong>Ответ:</strong> Пересечение с осью $Oy$: $(0; 2)$. Пересечение с осью $Ox$: $x = \pm 2\arccos\left(\frac{1 - \sqrt{3}}{2}\right) + 4\pi n, n \in \mathbb{Z}$.</p><p><strong>4. Промежутки монотонности и точки экстремума</strong></p><p>Находим первую производную: $f'(x) = -2\sin\frac{x}{2} + 2\sin x$. Преобразуем ее: $f'(x) = -2\sin\frac{x}{2} + 4\sin\frac{x}{2}\cos\frac{x}{2} = 2\sin\frac{x}{2}(2\cos\frac{x}{2} - 1)$.</p><p>Критические точки находим из условия $f'(x)=0$:</p><p>1) $\sin\frac{x}{2}=0 \implies x=2\pi n, n \in \mathbb{Z}$.</p><p>2) $\cos\frac{x}{2}=\frac{1}{2} \implies x=\pm\frac{2\pi}{3}+4\pi k, k \in \mathbb{Z}$.</p><p>Анализ знака $f'(x)$ на периоде $[0, 4\pi]$ показывает, что функция возрастает на $(0, \frac{2\pi}{3})$ и $(2\pi, \frac{10\pi}{3})$, и убывает на $(\frac{2\pi}{3}, 2\pi)$ и $(\frac{10\pi}{3}, 4\pi)$.</p><p>Экстремумы:</p><p>$x_{max} = \pm \frac{2\pi}{3} + 4\pi n$, значение $f(\pm \frac{2\pi}{3}) = 3$.</p><p>$x_{min} = 4\pi n$, значение $f(0) = 2$.</p><p>$x_{min} = 2\pi + 4\pi n$, значение $f(2\pi) = -6$.</p><p><strong>Ответ:</strong> Функция возрастает на интервалах $(4\pi n, \frac{2\pi}{3}+4\pi n)$ и $(2\pi+4\pi n, \frac{10\pi}{3}+4\pi n)$. Убывает на интервалах $(\frac{2\pi}{3}+4\pi n, 2\pi+4\pi n)$ и $(\frac{10\pi}{3}+4\pi n, 4\pi(n+1))$. Точки максимума $x = \pm\frac{2\pi}{3}+4\pi n$ ($y_{max}=3$), точки минимума $x=4\pi n$ ($y_{min}=2$) и $x=2\pi+4\pi n$ ($y_{min}=-6$), для $n \in \mathbb{Z}$.</p><p><strong>5. Промежутки выпуклости, вогнутости и точки перегиба</strong></p><p>Находим вторую производную: $f''(x) = -\cos\frac{x}{2} + 2\cos x$.</p><p>Приравнивая к нулю, получаем $4\cos^2\frac{x}{2} - \cos\frac{x}{2} - 2 = 0$.</p><p>Корни уравнения: $\cos\frac{x}{2} = \frac{1 \pm \sqrt{33}}{8}$.</p><p>Обозначим $x_1 = 2\arccos\left(\frac{1+\sqrt{33}}{8}\right)$ и $x_2 = 2\arccos\left(\frac{1-\sqrt{33}}{8}\right)$.</p><p>Анализ знака $f''(x)$ показывает, что функция вогнутая (выпукла вниз), когда $f''(x) > 0$, и выпуклая (выпукла вверх), когда $f''(x) < 0$. Точки перегиба — это точки, где меняется знак второй производной.</p><p><strong>Ответ:</strong> Функция вогнутая на интервалах $(-x_1+4\pi n, x_1+4\pi n)$ и $(x_2+4\pi n, 4\pi-x_2+4\pi n)$. Выпуклая на интервалах $(x_1+4\pi n, x_2+4\pi n)$ и $(4\pi-x_2+4\pi n, 4\pi-x_1+4\pi n)$, где $x_1 = 2\arccos\left(\frac{1+\sqrt{33}}{8}\right)$, $x_2 = 2\arccos\left(\frac{1-\sqrt{33}}{8}\right)$, $n \in \mathbb{Z}$. Точки перегиба: $x=\pm x_1+4\pi n$ и $x=\pm x_2+4\pi n$.</p><p><strong>6. Область значений</strong></p><p>Сравнивая значения в точках экстремума, находим глобальные максимум и минимум.</p><p>Глобальный максимум: $y_{max}=3$.</p><p>Глобальный минимум: $y_{min}=-6$.</p><p><strong>Ответ:</strong> $E(f) = [-6; 3]$.</p><p><strong>7. Построение графика</strong></p><p>На основе проведенного анализа строим график функции. Отмечаем на координатной плоскости точки экстремумов, точки пересечения с осями. Соединяем их плавными линиями с учетом интервалов монотонности и выпуклости. 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<circle cx="-125.66" cy="-50" r="1.5" fill="red"></circle></svg></p><p><strong>Ответ:</strong> Исследование функции проведено, ее график построен.</p>" ] #original: array:6 [ "id" => 1552152 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116} "task" => array:2 [ "refs" => "1120431" "type" => "task" ] "text" => "<p>Проведем полное исследование функции $f(x) = 4\cos\frac{x}{2} - 2\cos x$ и построим ее график, выполнив следующие шаги.</p><p><strong>1. Область определения</strong></p><p>Функция состоит из комбинации тригонометрических функций $\cos\frac{x}{2}$ и $\cos x$, которые определены для всех действительных чисел. Следовательно, область определения функции $f(x)$ — это множество всех действительных чисел.</p><p><strong>Ответ:</strong> $D(f) = (-\infty; +\infty)$.</p><p><strong>2. Четность и периодичность</strong></p><p>Проверим функцию на четность: $f(-x) = 4\cos\left(-\frac{x}{2}\right) - 2\cos(-x) = 4\cos\frac{x}{2} - 2\cos x = f(x)$, так как функция косинуса является четной. Таким образом, функция $f(x)$ — четная, ее график симметричен относительно оси ординат ($Oy$).</p><p>Период для слагаемого $4\cos\frac{x}{2}$ равен $T_1 = \frac{2\pi}{1/2} = 4\pi$. Период для слагаемого $-2\cos x$ равен $T_2 = 2\pi$. Наименьший общий период функции $f(x)$ равен наименьшему общему кратному периодов $T_1$ и $T_2$, то есть $T = \text{НОК}(4\pi, 2\pi) = 4\pi$.</p><p><strong>Ответ:</strong> Функция является четной и периодической с основным периодом $T = 4\pi$.</p><p><strong>3. Точки пересечения с осями координат</strong></p><p>С осью $Oy$: при $x=0$ имеем $f(0) = 4\cos(0) - 2\cos(0) = 4 - 2 = 2$. Точка пересечения — $(0; 2)$.</p><p>С осью $Ox$: решаем уравнение $f(x) = 0$, то есть $4\cos\frac{x}{2} - 2\cos x = 0$. Используем формулу косинуса двойного угла $\cos x = 2\cos^2\frac{x}{2} - 1$:</p><p>$4\cos\frac{x}{2} - 2(2\cos^2\frac{x}{2} - 1) = 0$, что приводит к квадратному уравнению относительно $\cos\frac{x}{2}$: $2\cos^2\frac{x}{2} - 2\cos\frac{x}{2} - 1 = 0$.</p><p>Решая его, получаем $\cos\frac{x}{2} = \frac{1 \pm \sqrt{3}}{2}$. Так как $|\cos \alpha| \le 1$, подходит только корень $\cos\frac{x}{2} = \frac{1 - \sqrt{3}}{2}$.</p><p>Отсюда $x = \pm 2\arccos\left(\frac{1 - \sqrt{3}}{2}\right) + 4\pi n$, где $n \in \mathbb{Z}$.</p><p><strong>Ответ:</strong> Пересечение с осью $Oy$: $(0; 2)$. Пересечение с осью $Ox$: $x = \pm 2\arccos\left(\frac{1 - \sqrt{3}}{2}\right) + 4\pi n, n \in \mathbb{Z}$.</p><p><strong>4. Промежутки монотонности и точки экстремума</strong></p><p>Находим первую производную: $f'(x) = -2\sin\frac{x}{2} + 2\sin x$. Преобразуем ее: $f'(x) = -2\sin\frac{x}{2} + 4\sin\frac{x}{2}\cos\frac{x}{2} = 2\sin\frac{x}{2}(2\cos\frac{x}{2} - 1)$.</p><p>Критические точки находим из условия $f'(x)=0$:</p><p>1) $\sin\frac{x}{2}=0 \implies x=2\pi n, n \in \mathbb{Z}$.</p><p>2) $\cos\frac{x}{2}=\frac{1}{2} \implies x=\pm\frac{2\pi}{3}+4\pi k, k \in \mathbb{Z}$.</p><p>Анализ знака $f'(x)$ на периоде $[0, 4\pi]$ показывает, что функция возрастает на $(0, \frac{2\pi}{3})$ и $(2\pi, \frac{10\pi}{3})$, и убывает на $(\frac{2\pi}{3}, 2\pi)$ и $(\frac{10\pi}{3}, 4\pi)$.</p><p>Экстремумы:</p><p>$x_{max} = \pm \frac{2\pi}{3} + 4\pi n$, значение $f(\pm \frac{2\pi}{3}) = 3$.</p><p>$x_{min} = 4\pi n$, значение $f(0) = 2$.</p><p>$x_{min} = 2\pi + 4\pi n$, значение $f(2\pi) = -6$.</p><p><strong>Ответ:</strong> Функция возрастает на интервалах $(4\pi n, \frac{2\pi}{3}+4\pi n)$ и $(2\pi+4\pi n, \frac{10\pi}{3}+4\pi n)$. Убывает на интервалах $(\frac{2\pi}{3}+4\pi n, 2\pi+4\pi n)$ и $(\frac{10\pi}{3}+4\pi n, 4\pi(n+1))$. Точки максимума $x = \pm\frac{2\pi}{3}+4\pi n$ ($y_{max}=3$), точки минимума $x=4\pi n$ ($y_{min}=2$) и $x=2\pi+4\pi n$ ($y_{min}=-6$), для $n \in \mathbb{Z}$.</p><p><strong>5. Промежутки выпуклости, вогнутости и точки перегиба</strong></p><p>Находим вторую производную: $f''(x) = -\cos\frac{x}{2} + 2\cos x$.</p><p>Приравнивая к нулю, получаем $4\cos^2\frac{x}{2} - \cos\frac{x}{2} - 2 = 0$.</p><p>Корни уравнения: $\cos\frac{x}{2} = \frac{1 \pm \sqrt{33}}{8}$.</p><p>Обозначим $x_1 = 2\arccos\left(\frac{1+\sqrt{33}}{8}\right)$ и $x_2 = 2\arccos\left(\frac{1-\sqrt{33}}{8}\right)$.</p><p>Анализ знака $f''(x)$ показывает, что функция вогнутая (выпукла вниз), когда $f''(x) > 0$, и выпуклая (выпукла вверх), когда $f''(x) < 0$. Точки перегиба — это точки, где меняется знак второй производной.</p><p><strong>Ответ:</strong> Функция вогнутая на интервалах $(-x_1+4\pi n, x_1+4\pi n)$ и $(x_2+4\pi n, 4\pi-x_2+4\pi n)$. Выпуклая на интервалах $(x_1+4\pi n, x_2+4\pi n)$ и $(4\pi-x_2+4\pi n, 4\pi-x_1+4\pi n)$, где $x_1 = 2\arccos\left(\frac{1+\sqrt{33}}{8}\right)$, $x_2 = 2\arccos\left(\frac{1-\sqrt{33}}{8}\right)$, $n \in \mathbb{Z}$. Точки перегиба: $x=\pm x_1+4\pi n$ и $x=\pm x_2+4\pi n$.</p><p><strong>6. Область значений</strong></p><p>Сравнивая значения в точках экстремума, находим глобальные максимум и минимум.</p><p>Глобальный максимум: $y_{max}=3$.</p><p>Глобальный минимум: $y_{min}=-6$.</p><p><strong>Ответ:</strong> $E(f) = [-6; 3]$.</p><p><strong>7. Построение графика</strong></p><p>На основе проведенного анализа строим график функции. Отмечаем на координатной плоскости точки экстремумов, точки пересечения с осями. Соединяем их плавными линиями с учетом интервалов монотонности и выпуклости. 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<circle cx="-125.66" cy="-50" r="1.5" fill="red"></circle></svg></p><p><strong>Ответ:</strong> Исследование функции проведено, ее график построен.</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" => "1120432" "type" => "task" ] "previous" => array:2 [ "refs" => "1120430" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1135 #items: array:1 [ 0 => App\Models\Book {#1157} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1141 #items: array:1 [ 0 => App\Models\BookPage {#1108 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1099280 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "127" "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/page-2-127" "field_display_title" => "127" "field_folder" => "2" "field_image_name" => "127" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1118 #items: array:1 [ 0 => App\Models\Branch {#1154} ] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1117 #items: array:1 [ 0 => App\Models\Book {#1157} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1113 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1106 #items: array:1 [ 0 => App\Models\Element {#1105 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1262705 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1070 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1262705 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1070 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1099281" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1099279" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1344 #items: array:8 [ 0 => App\Models\Task {#1358 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1120429 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "126" "field_page_end" => "127" "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-5-2zad-6" "field_display_title" => "6" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1359 …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 {#1360 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1361 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1362 …2} "content" => Illuminate\Database\Eloquent\Collection {#1364 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1370 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1120429 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "126" "field_page_end" => "127" "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-5-2zad-6" "field_display_title" => "6" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1359 …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 {#1360 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1361 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1362 …2} "content" => Illuminate\Database\Eloquent\Collection {#1364 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1370 …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 {#1365 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1120430 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "127" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-5-2zad-7" "field_display_title" => "7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1363 …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 {#1372 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1368 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1366 …2} "content" => Illuminate\Database\Eloquent\Collection {#1367 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1382 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1120430 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "127" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-5-2zad-7" "field_display_title" => "7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1363 …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 {#1372 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1368 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1366 …2} "content" => Illuminate\Database\Eloquent\Collection {#1367 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1382 …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 {#1377 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1120431 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "127" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-5-2zad-8" "field_display_title" => "8" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1369 …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 {#1384 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1380 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1378 …2} "content" => Illuminate\Database\Eloquent\Collection {#1381 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1379 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1120431 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "127" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-5-2zad-8" "field_display_title" => "8" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1369 …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 {#1384 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1380 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1378 …2} "content" => Illuminate\Database\Eloquent\Collection {#1381 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1379 …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 {#1389 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1120432 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "127" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-5-2zad-9" "field_display_title" => "9" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1390 …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 {#1391 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1392 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1393 …2} "content" => Illuminate\Database\Eloquent\Collection {#1395 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1401 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1120432 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "127" "field_page_end" => null …19 ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 4 => App\Models\Task {#1396 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 5 => App\Models\Task {#1408 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 6 => App\Models\Task {#1420 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 7 => App\Models\Task {#1432 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1099280 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "127" "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/page-2-127" 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№8 (с. 127)
Условие. №8 (с. 127)
скриншот условия
8. (4) Исследуйте функцию $f(x) = 4\cos\frac{x}{2} - 2\cos x$ и постройте ее график.
9. (2) Дана функция $g(x) = 2\sin x$ определенная на промежутке
Решение 2 (rus). №8 (с. 127)
Проведем полное исследование функции $f(x) = 4\cos\frac{x}{2} - 2\cos x$ и построим ее график, выполнив следующие шаги.
1. Область определения
Функция состоит из комбинации тригонометрических функций $\cos\frac{x}{2}$ и $\cos x$, которые определены для всех действительных чисел. Следовательно, область определения функции $f(x)$ — это множество всех действительных чисел.
Ответ: $D(f) = (-\infty; +\infty)$.
2. Четность и периодичность
Проверим функцию на четность: $f(-x) = 4\cos\left(-\frac{x}{2}\right) - 2\cos(-x) = 4\cos\frac{x}{2} - 2\cos x = f(x)$, так как функция косинуса является четной. Таким образом, функция $f(x)$ — четная, ее график симметричен относительно оси ординат ($Oy$).
Период для слагаемого $4\cos\frac{x}{2}$ равен $T_1 = \frac{2\pi}{1/2} = 4\pi$. Период для слагаемого $-2\cos x$ равен $T_2 = 2\pi$. Наименьший общий период функции $f(x)$ равен наименьшему общему кратному периодов $T_1$ и $T_2$, то есть $T = \text{НОК}(4\pi, 2\pi) = 4\pi$.
Ответ: Функция является четной и периодической с основным периодом $T = 4\pi$.
3. Точки пересечения с осями координат
С осью $Oy$: при $x=0$ имеем $f(0) = 4\cos(0) - 2\cos(0) = 4 - 2 = 2$. Точка пересечения — $(0; 2)$.
С осью $Ox$: решаем уравнение $f(x) = 0$, то есть $4\cos\frac{x}{2} - 2\cos x = 0$. Используем формулу косинуса двойного угла $\cos x = 2\cos^2\frac{x}{2} - 1$:
$4\cos\frac{x}{2} - 2(2\cos^2\frac{x}{2} - 1) = 0$, что приводит к квадратному уравнению относительно $\cos\frac{x}{2}$: $2\cos^2\frac{x}{2} - 2\cos\frac{x}{2} - 1 = 0$.
Решая его, получаем $\cos\frac{x}{2} = \frac{1 \pm \sqrt{3}}{2}$. Так как $|\cos \alpha| \le 1$, подходит только корень $\cos\frac{x}{2} = \frac{1 - \sqrt{3}}{2}$.
Отсюда $x = \pm 2\arccos\left(\frac{1 - \sqrt{3}}{2}\right) + 4\pi n$, где $n \in \mathbb{Z}$.
Ответ: Пересечение с осью $Oy$: $(0; 2)$. Пересечение с осью $Ox$: $x = \pm 2\arccos\left(\frac{1 - \sqrt{3}}{2}\right) + 4\pi n, n \in \mathbb{Z}$.
4. Промежутки монотонности и точки экстремума
Находим первую производную: $f'(x) = -2\sin\frac{x}{2} + 2\sin x$. Преобразуем ее: $f'(x) = -2\sin\frac{x}{2} + 4\sin\frac{x}{2}\cos\frac{x}{2} = 2\sin\frac{x}{2}(2\cos\frac{x}{2} - 1)$.
Критические точки находим из условия $f'(x)=0$:
1) $\sin\frac{x}{2}=0 \implies x=2\pi n, n \in \mathbb{Z}$.
2) $\cos\frac{x}{2}=\frac{1}{2} \implies x=\pm\frac{2\pi}{3}+4\pi k, k \in \mathbb{Z}$.
Анализ знака $f'(x)$ на периоде $[0, 4\pi]$ показывает, что функция возрастает на $(0, \frac{2\pi}{3})$ и $(2\pi, \frac{10\pi}{3})$, и убывает на $(\frac{2\pi}{3}, 2\pi)$ и $(\frac{10\pi}{3}, 4\pi)$.
Экстремумы:
$x_{max} = \pm \frac{2\pi}{3} + 4\pi n$, значение $f(\pm \frac{2\pi}{3}) = 3$.
$x_{min} = 4\pi n$, значение $f(0) = 2$.
$x_{min} = 2\pi + 4\pi n$, значение $f(2\pi) = -6$.
Ответ: Функция возрастает на интервалах $(4\pi n, \frac{2\pi}{3}+4\pi n)$ и $(2\pi+4\pi n, \frac{10\pi}{3}+4\pi n)$. Убывает на интервалах $(\frac{2\pi}{3}+4\pi n, 2\pi+4\pi n)$ и $(\frac{10\pi}{3}+4\pi n, 4\pi(n+1))$. Точки максимума $x = \pm\frac{2\pi}{3}+4\pi n$ ($y_{max}=3$), точки минимума $x=4\pi n$ ($y_{min}=2$) и $x=2\pi+4\pi n$ ($y_{min}=-6$), для $n \in \mathbb{Z}$.
5. Промежутки выпуклости, вогнутости и точки перегиба
Находим вторую производную: $f''(x) = -\cos\frac{x}{2} + 2\cos x$.
Приравнивая к нулю, получаем $4\cos^2\frac{x}{2} - \cos\frac{x}{2} - 2 = 0$.
Корни уравнения: $\cos\frac{x}{2} = \frac{1 \pm \sqrt{33}}{8}$.
Обозначим $x_1 = 2\arccos\left(\frac{1+\sqrt{33}}{8}\right)$ и $x_2 = 2\arccos\left(\frac{1-\sqrt{33}}{8}\right)$.
Анализ знака $f''(x)$ показывает, что функция вогнутая (выпукла вниз), когда $f''(x) > 0$, и выпуклая (выпукла вверх), когда $f''(x) < 0$. Точки перегиба — это точки, где меняется знак второй производной.
Ответ: Функция вогнутая на интервалах $(-x_1+4\pi n, x_1+4\pi n)$ и $(x_2+4\pi n, 4\pi-x_2+4\pi n)$. Выпуклая на интервалах $(x_1+4\pi n, x_2+4\pi n)$ и $(4\pi-x_2+4\pi n, 4\pi-x_1+4\pi n)$, где $x_1 = 2\arccos\left(\frac{1+\sqrt{33}}{8}\right)$, $x_2 = 2\arccos\left(\frac{1-\sqrt{33}}{8}\right)$, $n \in \mathbb{Z}$. Точки перегиба: $x=\pm x_1+4\pi n$ и $x=\pm x_2+4\pi n$.
6. Область значений
Сравнивая значения в точках экстремума, находим глобальные максимум и минимум.
Глобальный максимум: $y_{max}=3$.
Глобальный минимум: $y_{min}=-6$.
Ответ: $E(f) = [-6; 3]$.
7. Построение графика
На основе проведенного анализа строим график функции. Отмечаем на координатной плоскости точки экстремумов, точки пересечения с осями. Соединяем их плавными линиями с учетом интервалов монотонности и выпуклости. Затем, используя свойство периодичности, повторяем построенный фрагмент графика вдоль всей оси $Ox$ с периодом $4\pi$.
Ответ: Исследование функции проведено, ее график построен.
Другие задания:
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