Let $X$ be a [normed vector space](/page/Normed%20Vector%20Space), let $\mathcal A\subset X$, and let $I:\mathcal A\to\mathbb R$ be a functional. Suppose that $u_*\in\mathcal A$ satisfies
paragraph
admin
\begin{align*}
I[u_*]\le I[w]\quad\text{for every }w\in\mathcal A.
\end{align*}
latex_env
admin
Let $v\in X$ and suppose there exists $\varepsilon_0>0$ such that $u_*+\varepsilon v\in\mathcal A$ for every $\varepsilon\in(-\varepsilon_0,\varepsilon_0)$. If the map $\varepsilon\mapsto I[u_*+\varepsilon v]$ is differentiable at $0$, then