What is the extreme f(x,y)=x^3+xy^2+6xy ?

The extreme f(x,y)=x^3+xy^2+6xy is Saddle(0,0),Saddle(0,-6),Minimum(sqrt(3),-3),Maximum(-sqrt(3),-3)

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extreme f(x,y)=x^3+xy^2+6xy

extreme

Saddle (0, 0), Saddle (0, - 6), Minimum (3, - 3), Maximum (- 3, - 3)

Solution steps

Find the critical points:

(0, 0), (0, - 6), (3, - 3), (- 3, - 3)

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

D (x, y) = 12 x^{2} - 4 y^{2} - 24 y - 36

Check the sign of at each critical point

Check critical point

(0, 0) :

Saddle

Check critical point

(0, - 6) :

Saddle

Check critical point

(3, - 3) :

Minimum

Check critical point

(- 3, - 3) :

Maximum

Saddle (0, 0), Saddle (0, - 6), Minimum (3, - 3), Maximum (- 3, - 3)

What is the extreme f(x,y)=x^3+xy^2+6xy ?
The extreme f(x,y)=x^3+xy^2+6xy is Saddle(0,0),Saddle(0,-6),Minimum(sqrt(3),-3),Maximum(-sqrt(3),-3)