What is the extreme f(x,y)=9x^2+2y^2-xy^2 ?

The extreme f(x,y)=9x^2+2y^2-xy^2 is Minimum(0,0),Saddle(2,6),Saddle(2,-6)

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extreme f(x,y)=9x^2+2y^2-xy^2

extreme points

f (x, y) = 9 x^{2} + 2 y^{2} - x y^{2}

Minimum (0, 0), Saddle (2, 6), Saddle (2, - 6)

Solution steps

Find the critical points:

(0, 0), (2, 6), (2, - 6)

\frac{\partial ^{2} f}{\partial x ^{2}} = 4 - 2 x

D (x, y) = 18 (4 - 2 x) - 4 y^{2}

Check the sign of at each critical point

Check critical point

(0, 0) :

Minimum

Check critical point

(2, 6) :

Saddle

Check critical point

(2, - 6) :

Saddle

Minimum (0, 0), Saddle (2, 6), Saddle (2, - 6)

What is the extreme f(x,y)=9x^2+2y^2-xy^2 ?
The extreme f(x,y)=9x^2+2y^2-xy^2 is Minimum(0,0),Saddle(2,6),Saddle(2,-6)