$a$ is a stationary point of a scalar field $f$ whose Hessian at $a$ has two opposite sign diagonal entries. Is $a$ a saddle point?

Question

Let $S$ be some subset of $\mathbb R^n$ , $f:S \to \mathbb R$ is a function such that for some $a \in S$ , there is an open ball $B(a)\subseteq S$ such that $f$ has all second order partial derivatives (continuous) in $B(a)$ , $\nabla f(a)=O $ and the Hessian of $f$ at $a$ has at least two opposite sign diagonal entries . Then how to prove that $a$ is a saddle point of $f$ ?

Answer 1

Assume that $p:=f_{.11}(a)>0$. Then by Taylor's theorem for one variable $$f(a+te_1)=f(a)+{p\over2} t^2+o(|t|^2)=f(a)+\left({p\over2}+o(1)\right)t^2\qquad(t\to 0)\ .$$ It follows that for all sufficiently small $|t|>0$ one has $$f(a+te_1)>f(a)\ .$$ Similarly, if $f_{.22}(a)=:-p<0$ then $$f(a+te_2)<f(a)$$ for all sufficiently small $|t|$.

Therefore we can conclude that under the circumstances described in the question the function $f$ has neither a local maximum nor a local maximum at $a$.