Let $a, b \in \mathbb{R}^n$ be linearly independent, |a| = 5, |b| = 10. Functions $f_a, f_b$ on the sphere $S_1(0) = ${$x : |x| = 1$}$ \subset \mathbb{R}^n $ are defined as follows: $f_a(x)$ is the angular diameter of the sphere $S_1(a) = ${$y : |y - a| = 1$} viewed from x, similarly, $f_b(x)$ is the angular diameter of $S_1(b)$ from x.
Prove that every point of local extremum of the function $h=f_a + f_b$ on $S_1(0)$ is some linear combination of $a, b$.
so I tried this:
I showed that $sin(\frac{1}{2}f_a)=1/|x-a| \rightarrow h(x)=2arcsin(\frac{1}{|x-a|})+2arcsin(\frac{1}{|x-b|})$ and then trying to use the gradient to prove it, but I really don't know what to do from here.