Second derivative test in several variables

Remarks

For $n = 2$, let $D = f_{xx}(a) f_{yy}(a) - f_{xy}(a)^2$ be the determinant of $H$. If $D > 0$ and $f_{xx}(a) > 0$, then $a$ is a local minimum. If $D > 0$ and $f_{xx}(a) < 0$, then $a$ is a local maximum. If $D < 0$, then $a$ is a saddle point. The proof uses the second-order Taylor expansion. If $H$ is positive definite, then $h^{\mathsf{T}} H h \ge m \|h\|^2$ for some $m > 0$, and this term dominates the remainder. The functions $x^4 + y^4$, $-(x^4 + y^4)$ and $x^4 - y^4$ all have zero Hessian at the origin. The origin is a minimum, a maximum and a saddle point, respectively, so the test cannot decide when $H$ is singular and semidefinite. By Sylvester's criterion, $H$ is positive definite if and only if all its leading principal minors are positive.