Let $\Omega\subseteq \mathbb{R}^n$ be an open set and let be $f:\Omega \to \mathbb{R}$ a real values multivariable function. Suppose that $f\in \mathcal{C}^2(\Omega)$.
I can define the gradient of $f$:
$$\nabla f=(\partial_1 f, \partial_2 f, ...., \partial_n f)$$
and the hessian matrix.
Let $P\in \Omega$ such that $\nabla f (P)=0$, in other words, $P$ is a critical point of $f$.
Now, I was wondering myself if the classification criterion using the Hessian matrix signature is an "if and only if" ($ \iff$)? More precisely, if the hessian matrix is, for example, positive definite when it is evaluated in $P$, then $P$ is a local maximum point (sharp) of the function $f$. On the other hand, is the vice-versa true? That is, if $P$ is a local maximum point of $f$, then the hessian is positive definite or I can only say that the hessian is semi positive definite, and, in this second case, why?
Can I repeat the same argument assuming that the Hessian matrix is semi definite positive in $P$?