Critical Points and the Hessian — Cheat sheet

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Core rule

At a stationary point: $D=f_{xx}f_{yy}-f_{xy}^2$ . $D<0$ : saddle. $D>0$ : use the sign of $f_{xx}$ . $D=0$ : inconclusive.

This is a local test requiring second-order regularity. Semidefinite Hessians do not settle the classification.