Higher-order test for separated extrema
= 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.