What is the extreme f(x,y)=x^3+3xy^2-15x-12y ?

The extreme f(x,y)=x^3+3xy^2-15x-12y is Saddle(1,2),Saddle(-1,-2),Minimum(2,1),Maximum(-2,-1)

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extreme f(x,y)=x^3+3xy^2-15x-12y

extreme points

f (x, y) = x^{3} + 3 x y^{2} - 15 x - 12 y

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

Solution steps

Find the critical points:

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

\frac{\partial ^{2} f}{\partial x ^{2}} = 6 x

D (x, y) = 36 x^{2} - 36 y^{2}

Check the sign of at each critical point

Check critical point

(1, 2) :

Saddle

Check critical point

(- 1, - 2) :

Saddle

Check critical point

(2, 1) :

Minimum

Check critical point

(- 2, - 1) :

Maximum

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

What is the extreme f(x,y)=x^3+3xy^2-15x-12y ?
The extreme f(x,y)=x^3+3xy^2-15x-12y is Saddle(1,2),Saddle(-1,-2),Minimum(2,1),Maximum(-2,-1)