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**syme.gabriel** Let $\displaystyle f(x)$ be a $\displaystyle \mathcal{C}^1$ function and let $\displaystyle M$ be a subspace of $\displaystyle \mathbb{R}^n$. Suppose $\displaystyle x^\ast \in M$ minimizes $\displaystyle f(x)$ on $\displaystyle M$. Show $\displaystyle \nabla f(x^\ast) \in M^\perp$.

My only idea is to take any $\displaystyle x \in M$ and try to show something with the function $\displaystyle \varphi(t) = f(x^\ast + tx)$, but I haven't been able to get anywhere with that. Any help is greatly appreciated!