$C^1$ function orthogonality implies maximum

Question

I'm trying to figure out this problem that I found in a Ph.d. entry test. It's probably taken out of Lima's analysis books. So far I have nothing so any hints would be great.

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Let $F: \mathbb{R}^n \setminus 0 \to \mathbb{R}^n$ be a $C^1$ function such that $\nabla f(x)$ and $x$ are orthogonal for all $x.$

Prove that $F$ attains its maximum.

Answer 1

Look at the function $g(t)=f(tx)$. Compute its derivative.

$tg'(t)=\nabla f(tx)\cdot tx=0$.

Therefore $g$ is constant. That means that $f$ is constant on each ray.

This means that the values that $f$ takes are exactly those that it takes on the unit circle, which is compact.