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Задать вопрос экспертам

Консультация № 158000

17.01.2009, 23:57

0.00 руб.

0 1 1

Образование

Здравствуйте уважаемые эксперты, помогите решить задачи.
1. Найти экстремумы функции z=1/2xy+(47-x-y)(x/3+y/4).
2. Вычислить площадь фигуры, ограниченной линиями xy=4 и x+y=5

Обсуждение

Неизвестный

19.01.2009, 01:19

общий

это ответ

Здравствуйте, Lukovsky Sergey !

отвечая на вопрос №1...
находим первые производные по х и у:
dz/dx=-y/12-2*x/3+47/3
dz/dy=-x/12-y/2+47/4

далее находим стационарные точки, т.е. приравниваем к нули обе полученные производные.
получаем одну точку (21;20)

теперь находим вторые производные: по х дважды, по у дважды и по ху.
Итак:
d2z/dx2=-2/3
d2z/dy2=-1/2
d2z/dxdy=-1/12

Далее:

(-2/3)*(-1/2)-(-1/12)^2=47/144

что больше 0. И так как вторая производная z по х меньше нуля, то имеем локальныйй максимум в нашей стационарной точке.

отвечая на вопрос №2.

находим точки пересечения...
это 2 точки: х1=1; у1=4; х2=4; у2=1.

искомая площадь будет равна

интеграл от 1 до 4 от (5-x - 4/x)dx=15/2-8ln(2) или примерно 1,955.

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