= Solution
The <gradient> vanishes exactly when $x^2/4-1=0$ and $y^2/4-1=0$, giving the four points $(\pm2,\pm2)$. The <Hessian matrix> is $\operatorname{diag}(x/2,y/2)$. It is positive definite at $(2,2)$, negative definite at $(-2,-2)$, and indefinite at the two mixed-sign points. Thus \b[$(2,2)$ is a strict local minimum, $(-2,-2)$ a strict local maximum, and $(2,-2),(-2,2)$ are <saddle points>] by the <second-derivative test>. Their respective <function> values are $-8/3$, $8/3$, and zero at both <saddle points>.
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