What is the extreme f(x,y)=x^2y+4xy-y^2+3 ?

The extreme f(x,y)=x^2y+4xy-y^2+3 is Saddle(0,0),Saddle(-4,0),Maximum(-2,-2)

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extreme f(x,y)=x^2y+4xy-y^2+3

extreme points

f (x, y) = x^{2} y + 4 x y - y^{2} + 3

Saddle (0, 0), Saddle (- 4, 0), Maximum (- 2, - 2)

Solution steps

Find the critical points:

(0, 0), (- 4, 0), (- 2, - 2)

\frac{\partial ^{2} f}{\partial x ^{2}} = - 2

D (x, y) = - 4 y - 4 x^{2} - 16 x - 16

Check the sign of at each critical point

Check critical point

(0, 0) :

Saddle

Check critical point

(- 4, 0) :

Saddle

Check critical point

(- 2, - 2) :

Maximum

Saddle (0, 0), Saddle (- 4, 0), Maximum (- 2, - 2)

What is the extreme f(x,y)=x^2y+4xy-y^2+3 ?
The extreme f(x,y)=x^2y+4xy-y^2+3 is Saddle(0,0),Saddle(-4,0),Maximum(-2,-2)