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

The extreme f(x,y)=x^2+y^2+x^2y^2 is Minimum(0,0)

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extreme f(x,y)=x^2+y^2+x^2y^2

extreme points

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

Minimum (0, 0)

Solution steps

Find the critical points:

(0, 0)

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

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

Check the sign of at each critical point

Check critical point

(0, 0) :

Minimum

Minimum (0, 0)

f (x) = x^{\frac{8}{9}} - 3

f (x) = x^{3} - 3 x^{2} + 3 x + 7

f (x) = (t - 4 t^{2})^{\frac{1}{3}}

f (x, y) = 1 - (x^{2} + y^{2})

\frac{x}{( x ^{2} + 1 )}

What is the extreme f(x,y)=x^2+y^2+x^2y^2 ?
The extreme f(x,y)=x^2+y^2+x^2y^2 is Minimum(0,0)