If $p\in R^{m}$ is a local minimum of $F:R^{m}\rightarrow R$, then can we conclude that $\dfrac{\partial ^2F}{\partial x \partial x'}[p]$ is positive definite?
I guess you guys answers have concluded that $\dfrac{\partial ^2F}{\partial x \partial x'}[p]$ can be only positive semi-definite.
How can I prove this result then?
Best Answer
A nontrivial counterexample is the function $f(x,y) = x^4+y^4$.
Clearly, $f$ has a local minimum at $(x,y) = (0,0)$.
However, the Hessian at $(0,0)$ is the zero-matrix, which is not positive definite.