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Let $D$ be a convex set in $\mathbb{R}^n$ and $f: D \to \mathbb{R}$ a concave and $C^1$ function. How do I show that $x^*$ is a global maximum for $f$ if and only if $f^{(1)}(x^*)y \leq 0$ for all $y$ pointing into $D$ at $x^*$ (Here $f^{(1)}$ denotes the first derivative of $f$)

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