= Solution
Evaluate the <Hessian matrix> at the five <stationary points>. At the origin it is
$$
H(0,0)=\begin{pmatrix}-2&-4\\-4&-2\end{pmatrix},
$$
whose <eigenvalues> are $-6$ and $2$, so the origin is a <saddle point>. At the two points with $x=y=\pm\sqrt3/2$,
$$
H=e^{-3/2}\begin{pmatrix}10&2\\2&10\end{pmatrix},
$$
whose <eigenvalues> are $12e^{-3/2}$ and $8e^{-3/2}$. Both are positive, so these are strict <local minima>. At the two points with $x=-y=\pm1/2$,
$$
H=e^{-1/2}\begin{pmatrix}-6&-2\\-2&-6\end{pmatrix},
$$
whose <eigenvalues> are $-8e^{-1/2}$ and $-4e^{-1/2}$. Both are negative, so these are strict <local maxima>. In summary,
$$
\boxed{(0,0):\text{ saddle};\quad x=y=\pm\sqrt3/2:\text{ minima};\quad x=-y=\pm1/2:\text{ maxima}.}
$$
The quadratic changes along the <orthogonal> directions $(1,1)$ and $(1,-1)$ also provide a direct justification of these classifications.
Back to article page
