I have a differentiable function $f:M \to \mathbb{R}$ where $M$ is a smooth manifold. If $p \in M $ is a point of local maxima, that is I have an open set $V \subset M$, $p \in V$, so that $f(p)\geq f(x), \forall x \in V$ , I need to show that $f_*|_p (v) = 0, \forall v \in T_p(M)$.
I tried using a chart $(U, \phi)$ around $p$ containing $V$, and then the function $f \small \;o \; \phi^{-1}:\mathbb{R}^n \to \mathbb{R}$, of course the domain being more specific. Then I tried to use the fact that this map gives a zero map as the derivative at $p$, but I am not sure whether I am on the right track. Any help is appreciated, thanks.
Recall that $(f_*)_p(X_p)=X_p(f)$, where we treat $X_p\in T_pM$ as a derivation on the ring of smooth functions. Therefore, showing that $(f_*)_p(X_p)=0$ whenever $f$ reaches a local maximum is the same as showing that the directional derivative of $f$ in any given direction is $0$ at $p$. Think you can take it from there?