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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)

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  • $\begingroup$ $\gamma$ does not exist if (e.g.) $x_0$ is not tangential to $M_0$. In general, $M_0$ might not have a smooth structure. $\endgroup$ – Arctic Char Aug 6 at 10:04
  • $\begingroup$ @ArcticChar Please take note of my edit. $\endgroup$ – 0xbadf00d Aug 6 at 12:37

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