What is the extreme f(x,y)=y^3-yx^2-3y^2*x^2+3x^4 ?

The extreme f(x,y)=y^3-yx^2-3y^2*x^2+3x^4 is

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extreme f(x,y)=y^3-yx^2-3y^2*x^2+3x^4

extreme

Saddle (\frac{1}{3}, \frac{1}{3}), Saddle (- \frac{1}{3}, \frac{1}{3}), Minimum (\frac{1}{2 6}, \frac{1}{6}), Minimum (- \frac{1}{2 6}, \frac{1}{6})

Solution steps

Find the critical points:

(\frac{1}{3}, \frac{1}{3}), (- \frac{1}{3}, \frac{1}{3}), (\frac{1}{2 6}, \frac{1}{6}), (- \frac{1}{2 6}, \frac{1}{6})

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

D (x, y) = - 216 x^{4} - 108 x^{2} y^{2} + 180 x^{2} y - 4 x^{2} - 36 y^{3} - 12 y^{2}

Check the sign of at each critical point

Check critical point

(\frac{1}{3}, \frac{1}{3}) :

Saddle

Check critical point

(- \frac{1}{3}, \frac{1}{3}) :

Saddle

Check critical point

(\frac{1}{2 6}, \frac{1}{6}) :

Minimum

Check critical point

(- \frac{1}{2 6}, \frac{1}{6}) :

Minimum

Saddle (\frac{1}{3}, \frac{1}{3}), Saddle (- \frac{1}{3}, \frac{1}{3}), Minimum (\frac{1}{2 6}, \frac{1}{6}), Minimum (- \frac{1}{2 6}, \frac{1}{6})

What is the extreme f(x,y)=y^3-yx^2-3y^2*x^2+3x^4 ?
The extreme f(x,y)=y^3-yx^2-3y^2*x^2+3x^4 is