differentiable map on sphere

Question

I'm trying to show that if $f:S^n\to \mathbb{R}$ is differentiable, then there are two distinct points $p,q\in S^n$ where the differentials $T_pf$ and $T_qf$ vanish. Any suggestions?

Answer 1

Well, since $S^n$ is compact and connected, its image under $f$ is compact, connected, hence an interval. So $f$ takes on a max and a min value. Either these occur at the same point, in which case $f$ is constant, so the differentials vanish everywhere, or they don't, in which case the differential vanishes at any point in the preimage of each of them (hence at least two points).