# $\nabla_y\nabla_vf(x^*)\geq0$ for any concave $f$ if and only if $y=-v$

$$f:\mathbb R^3\to\mathbb R$$ 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.