= Higher-order test for separated extrema
For $f(x,y)=g(x)+h(y)$, a degenerate <Hessian matrix> can be supplemented by the first nonzero term of each one-variable displacement. Two negative leading even powers give a strict <local maximum>, two positive ones a strict <local minimum>, and opposite signs give a saddle. Looking along the coordinate directions and bounding the separated terms avoids an inconclusive quadratic test.
For an explicit contrast, consider
$$
f_+(x,y)=x^4+y^4,
\qquad
f_-(x,y)=x^4-y^4.
$$
Both have zero <gradient> and the same zero <Hessian matrix> at the origin:
$$
H_{f_+}(0,0)=H_{f_-}(0,0)=\begin{pmatrix}0&0\\0&0\end{pmatrix}.
$$
Nevertheless, $f_+$ is positive at every nonzero point, so the origin is a strict <global minimum>. For $f_-$, the restrictions $f_-(t,0)=t^4>0$ and $f_-(0,t)=-t^4<0$ for $t\ne0$ show that the origin is a <saddle point of a scalar function>. Identical quadratic information can conceal different geometry when higher-order terms are decisive.
\Image[/hessian-degenerate-critical-points.png]
{title=The same zero Hessian can occur at a strict minimum or a saddle}
{description=The quartic functions $x^4+y^4$ and $x^4-y^4$ both have zero second <partial derivatives> at the origin. Their contour geometry and signs differ, so a singular <Hessian matrix> alone cannot classify the <critical point>.}
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