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Консультация № 194097
05.12.2018, 10:26
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Уважаемые эксперты! Пожалуйста, ответьте на вопрос:
Задания по теме 2.2.«Пределы».
11. а) ; б) ;
Задания по теме 2.3.«Производная»
I. Найти производную функции :
11. а) ; б) .
II. Найти вторую производную функции :
11. ;
III. Вычислить приближенно с помощью дифференциала:
11. ;
Задание по теме 2.4.«Исследование функций»
Провести полное исследование функции и построить её график.
11. а) ; б) ;
Задания по теме 2.5.«Функции нескольких переменных»
I. Найти стационарные точки функции и исследовать их на экстремум.
11. z=3x3+2xy2+x2 +4y2
II. Найти условный экстремум функции.
11. u=7x+2y+3z при 9x2+y2+z2 =36.
Задания по теме 2.2.«Пределы».
11. а) ; б) ;
Задания по теме 2.3.«Производная»
I. Найти производную функции :
11. а) ; б) .
II. Найти вторую производную функции :
11. ;
III. Вычислить приближенно с помощью дифференциала:
11. ;
Задание по теме 2.4.«Исследование функций»
Провести полное исследование функции и построить её график.
11. а) ; б) ;
Задания по теме 2.5.«Функции нескольких переменных»
I. Найти стационарные точки функции и исследовать их на экстремум.
11. z=3x3+2xy2+x2 +4y2
II. Найти условный экстремум функции.
11. u=7x+2y+3z при 9x2+y2+z2 =36.
Обсуждение
09.12.2018, 13:53
общий
это ответ
Здравствуйте, stroitel!
Представим уравнение связи в виде
и составим функцию 
Лагранжа; получим
Вычислим частные производные функции
получим
Приравняем частные производные функции к нулю, к трём полученным уравнениям присоединим уравнение связи. Тогда
Если
то
Следовательно,
-- стационарная точка функции 
Если
то
Следовательно,
-- стационарная точка функции
Проверим выполнение достаточного условия экстремума. Составим дифференциал второго порядка
Вычислим частные производные второго порядка; получим
Понятно, что в точке
дифференциал второго порядка 
а в точке 
дифференциал второго порядка 
Поэтому функция 
достигает в точке достигает условного максимума
а в точке -- условного минимума
Представим уравнение связи в виде
и составим функцию Лагранжа; получимВычислим частные производные функции
получимПриравняем частные производные функции
к нулю, к трём полученным уравнениям присоединим уравнение связи. ТогдаЕсли
тоСледовательно,
-- стационарная точка функции Если
тоСледовательно,
-- стационарная точка функции Проверим выполнение достаточного условия экстремума. Составим дифференциал второго порядка
Вычислим частные производные второго порядка; получим
Понятно, что в точке
дифференциал второго порядка а в точке дифференциал второго порядка Поэтому функция достигает в точке достигает условного максимумаа в точке
-- условного минимума
Об авторе:
Facta loquuntur.
Facta loquuntur.
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