Local minimum of an analytic function

Question

This is a follow-up to a previous question of mine. I know that any local minimum $x_0$ of a function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ has positive semi-definite Hessian $H(x_0) \succeq 0$. If $f$ is analytic, can we say moreover that there exists a neighbourhood $U$ of $x_0$ such that $$ H(x) \succeq 0 $$ for all $x \in U$? I don't see any reason for this being true since positive semi-definiteness is not a continuous property (while positive definiteness is), but can't find a counter-example.

Answer 1

Here is a very simple counter-example: $$f(x)=x^2y^2.$$ This gives $$H(x,y)=\begin{pmatrix} 2y^2&4xy\\ 4xy&2x^2 \end{pmatrix},$$ in particular $$H(x,x)=2x^2\begin{pmatrix} 1&2\\ 2&1 \end{pmatrix},$$ which is not positive semidefinite for $ x \ne 0$.