Extreme points

Find the extreme points for function
$ f(x,y,z)=x+y+z+x^2y^2+x^2z^2+y^2z^2 $
WE look for the stationary points. So , we have to solve the system

$$\frac{\partial f}{\partial x}=1+2x(y^2+ z^2)=0$$
$$\frac{\partial f}{\partial y}=1+2y(x^2+ z^2)=0$$
$$\frac{\partial f}{\partial z}=1+2z(x^2+ y^2)=0$$

We also have

$$x^2y-xy^2+z^2(y-x)=0$$

or

$$(x-y)(xy-z^2)=0$$

We also have

$$(x-z)(xz-y^2)=0$$
$$(y-z)(yz-x^2)=0$$

The only solutions is $ x=y=z=-\frac{1}{\sqrt[3]4} $
But this point is not an extreme point.

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