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Задать вопрос экспертам

Консультация № 161317

24.02.2009, 20:28

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Образование

Здраствуйте, уважаемые эксперты! Помогите пожалуйста. Нужно решить 2 уравнения:
1) Найти полный дифференциал функции z=f(x,y)
z=arcsin(1+x*y);
2) Найти экстремум функции двух переменных z=f(x,y)
z=2*x^2+3*x*y+y^3+x.

Заранее спасибо!

Обсуждение

Неизвестный

25.02.2009, 12:18

общий

это ответ

Здравствуйте, Бондаренко Кирилл Андреевич!
1) z=arcsin(1+xy)
dz=(dz/dx)*dx+(dz/dy)*dy
dz/dx, dz/dy - частные производные функции z (d- "круглые")
dz/dx=y/sqrt(1-(1+xy)²)=y/sqrt(-2xy-x²y²)
dz/dy=x/sqrt(1-(1+xy)²)=x/sqrt(-2xy-x²y²)
dz=y/sqrt(1-(1+xy)²)=y/sqrt(-2xy-x²y²) * dx + x/sqrt(1-(1+xy)²)=y/sqrt(-2xy-x²y²) * dy=
=(ydx+xdy)/sqrt(-2xy-x²y²)

2) z=2x²+3xy+y³+x
Найдем стационарные точки
dz/dx=4x+3y+1=0
dz/dy=3x+3y²=0

4*(-y²)+3y+1=0
x=-y²

4y²-3y-1=0
D=9+16=25
y₁=(3+5)/8=1
y₂=(3-5)/8=-1/4

x₁=-1²=-1
x₂=-(1/4)²=-1/16
Точки
М1(-1, 1)
М2(-1/16, -1/4)
A=d²z/dx²=4
B=d²z/dxdy=3
C=d²z/dy²=6y
S=AC-B²=24y-9
В точке М1 величина S=24-9=15>0, A=4>0, значит в точке М1 функция достигает локального минимума z_min=-1
В точке М2 величина S=-6-9=-15<0 - экстремма нет.

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