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

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

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

extreme

Saddle (0, 0), Saddle (1, 1), Minimum (\frac{1}{2}, \frac{1}{2}), Saddle (0, 0)

Solution steps

Find the critical points:

(0, 0), (1, 1), (\frac{1}{2}, \frac{1}{2}), (0, 0)

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

D (x, y) = - 12 x^{3} - 12 x^{2} y^{2} + 28 x y - 12 y^{3} - 1

Check the sign of at each critical point

Check critical point

(0, 0) :

Saddle

Check critical point

(1, 1) :

Saddle

Check critical point

(\frac{1}{2}, \frac{1}{2}) :

Minimum

Check critical point

(0, 0) :

Saddle

Saddle (0, 0), Saddle (1, 1), Minimum (\frac{1}{2}, \frac{1}{2}), Saddle (0, 0)

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