Let $X$ be a $\mathbb R$-Banach space, $M\subseteq X$ be open, $M_0\subseteq M$ be closed, $E\in C^1(M)$, $u\in M_0$ be an extremal point of $E$, $\varepsilon_0>0$ and $\gamma\in C^1(-\varepsilon_0,\varepsilon_0),M_0)$ with $\gamma(0)=u$ and $\gamma'(0)=x_0$ for some given $x_0\in X$.
Why can we conclude that $(E\circ\gamma)'(0)=0$? (And how can we show that $\gamma$ exists?)
Intuitively, it's clear to me that $0$ should be an extremal point of $E\circ\gamma$ and hence the claim holds true. The continuity should ensure that $E$ cannot jump along the curve away from the neighborhood of the local minimum/maximum $u$. But how can we prove this rigorously?
EDIT: You may assume that $M=\{\Phi=0\}$ for some $\Phi\in C^1(M,Y)$, $Y$ being another $\mathbb R$-Banach space. (I guess this implies that $M$ is some kind of Manifold)