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Задать вопрос экспертам

Консультация № 183946

31.08.2011, 18:11

65.00 руб.

31.08.2011, 18:30

0 2 2

Образование

Здравствуйте! Прошу помощи в следующем вопросе:

z=х²+y-15xy
1) Найти частные производные первого и второго порядка функции z(x,y).
2) Исследовать функцию на экстремум.
3) Исследовать функцию на условный экстремум при условии: х+у=1.

Обсуждение

давно

Асмик Гаряка

Профессор
230118
3054

31.08.2011, 19:24

общий

это ответ

Здравствуйте, Посетитель - 380343!

1)

2 )Приравняем частные производные к 0
2x-15y=0
1-15x=0
x=1/15
15y=2/15
y=2/225

AC-B^2=2*0-15^2=-225<0

Экстремума нет
3) Исследовать функцию на условный экстремум при условии: х+у=1.
Рассмотрим функцию Лагранжа F(x,y)+[$955$][$966$]=х^2+y-15xy+[$955$](x+y-1)
Найдем ее частные производные и приравняем их к 0
L'x=2x-15y+[$955$]=0
L'y=1-15x+[$955$]=0
L'[$955$]=x+y-1=0
y=1-x
2x-15(1-x)+[$955$]=17x-15+[$955$]=0
[$955$]=-17x+15
[$955$]=15x-1
15x-1=-17x+15
x=1/2
y=1/2
[$955$]=6.5
Значение z в этой точке равно 1/4+1/2-15/4=-3

давно

Орловский Дмитрий

Мастер-Эксперт
319965
1463

31.08.2011, 20:08

общий

это ответ

Здравствуйте, Посетитель - 380343!
Предыдущее решение третьей задачи частично ошибочно. Для исследования на экстремум нужно учитывать связь между дифференциалами. Из условия x+y=1 следует, что dx+dy=0. При этом предположении второй дифференциал
d²L=2(dx)²+2(dy)²-30dxdy=2(dx)²+2(dx)²+30(dx)²=34(dx)²
положительно определен. Следовательно, стационарная точка x=1/2,y=1/2 является точкой минимума.

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