Extreme points

Posted November 21st, 2007 by Structure

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

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

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

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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