Let $f(x,y,z)=(1 + |x|+|y|) \cosh(|x|-|y| + z^2)$.
$f(0,0,0)=1<f(x,y,z)$ for different $x,y,z$ so it is a global minimum.
Using $f(x,y,z)=f(|x|,|y|,|z|)$ I found there are no extrema at points outside of the axes, looking at the partial derivatives. How can I prove the same on the axes, except the origin?
Should I now consider (with the variables being positive, using symmetry) $f(x,y,0), f(x,0,y), f(0,y,z), f(x,0,0), f(0,y,0), f(0,0,z)$ and study their partial derivatives or is there a quicker method?