What is the extreme f(x,y)=5+3x^2+3y^2+2y^3+x^3 ?

The extreme f(x,y)=5+3x^2+3y^2+2y^3+x^3 is Minimum(0,0),Saddle(0,-1),Saddle(-2,0),Maximum(-2,-1)

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extreme f(x,y)=5+3x^2+3y^2+2y^3+x^3

extreme points

f (x, y) = 5 + 3 x^{2} + 3 y^{2} + 2 y^{3} + x^{3}

Minimum (0, 0), Saddle (0, - 1), Saddle (- 2, 0), Maximum (- 2, - 1)

Solution steps

Find the critical points:

(0, 0), (0, - 1), (- 2, 0), (- 2, - 1)

\frac{\partial ^{2} f}{\partial x ^{2}} = 12 y + 6

D (x, y) = (6 x + 6) (12 y + 6)

Check the sign of at each critical point

Check critical point

(0, 0) :

Minimum

Check critical point

(0, - 1) :

Saddle

Check critical point

(- 2, 0) :

Saddle

Check critical point

(- 2, - 1) :

Maximum

Minimum (0, 0), Saddle (0, - 1), Saddle (- 2, 0), Maximum (- 2, - 1)

What is the extreme f(x,y)=5+3x^2+3y^2+2y^3+x^3 ?
The extreme f(x,y)=5+3x^2+3y^2+2y^3+x^3 is Minimum(0,0),Saddle(0,-1),Saddle(-2,0),Maximum(-2,-1)