I'm having problems with this assignment:
$\begin{array}{rl} \min & x^3 + 2xyz - z^2 \\ \text{subject to} & x^2 + y^2 + z^2 \leq 1 \\ \end{array}$
Disregarding the constraint, find all points $(x,y,z)$ at which the gradient of the criterion function is zero. What is the criterion value at these points? Provide an argument why all these points are neither (unconstrained) local minima nor local maxima, but saddle points.
My approach: I calculated the gradient: $3 x^2+2 y z$; $2 x z$; $2 x y-2 z$ How can I find the values for $x,y$ and $z$ to let the elements be equal to zero?
Any help would be appreciated!