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$f:\mathbb R^3\to\mathbb R$ is an arbitrary concave function.

$H$ is a plane. $v$ is a given vector on $H$.

$x^*=\max_{x\in H} f(x)$

Prove that $\nabla_y\nabla_vf(x^*)\geq 0$ if and only if $y=-v$.


I draw a plot in 2D and the result is very intuitive. If we move the $x^*$ a little bit along $-v$, to $x^*-v\epsilon$, then we must have:

$$\nabla_vf(x^*-v\epsilon)\geq 0$$

since $f$ is concave. But idk how to formally prove the whole thing without a graphic illustration. Let me know if uploading a graph will help.

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