 Solution

For every stationary point found above, $\cos x_s\ne 0$ and hence $z_s\ne 0$. The Hessian matrix is diagonal, with entries $-(4x_s^2+3)z_s$ and $-2z_s$, soThe entries always have the same sign. ThereforeThe root on tangent branch $(n\pi -\pi /2,n\pi +\pi /2)$ has cosine sign $(-1{)}^n$. Thus even-numbered branches supply maxima and odd-numbered branches minima, symmetrically paired about the origin. There are no saddle points among these nondegenerate stationary points. The origin is a maximum with $z_s=1$, and is in fact the global maximum since $\cos x\le 1$ and $e^{-x^2-y^2}\le 1$.