= Solution
For a differentiable function at an interior <stationary point>, both first <partial derivatives> vanish. If the second derivatives are continuous, form the symmetric <Hessian matrix>
$$
H=\begin{pmatrix}f_{xx}&f_{xy}\\f_{xy}&f_{yy}\end{pmatrix},\qquad D=f_{xx}f_{yy}-f_{xy}^2.
$$
The <second-derivative test> gives a strict local minimum if $D>0$ and $f_{xx}>0$, a strict local maximum if $D>0$ and $f_{xx}<0$, and a <saddle point> if $D<0$. If $D=0$, higher-order terms must be examined. For example, $x^4+y^4$, $-x^4-y^4$ and $x^4-y^4$ all have zero <Hessian matrix> at the origin but respectively a minimum, maximum and saddle. These are local interior criteria; boundary extrema require separate treatment.
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