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I'm having trouble with a question:

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ a differentiable function. If $\dfrac{\partial f}{\partial u}(u)>0$ for all $u \in S^{n-1}$, there exists a $a\in \mathbb{R}^n$ such that $\dfrac{\partial f}{\partial v}(a)=0$, for all $v \in \mathbb{R}^n$.

Can anyone give me a hint, please?

TDg1
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    For people not used to your notation, you might explain that $\frac{\partial f}{\partial v}(a)$ is the directional derivative at $a$ in the direction of $v$ (not necessarily a unit vector). – Ted Shifrin May 27 '23 at 18:57

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HINT: The function must take on a minimum value somewhere inside the unit ball.

Ted Shifrin
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