$df$ vanish in a compact manifold in at least 2 points

Question

I need to prove that if $M$ is a compact manifold and $f$ is a smooth function in $M$, then $df$ vanish in at least 2 different points of $M$. I don't know where to start. Any suggestion will be welcome.

Answer 1

If $f$ is constant then $df=0$ everywhere. In general $df=0$ at least wherever $f$ achieves a local maximum or local minimum (and possibly elsewhere), if $f$ is nonconstant then by compactness the set of local extrema contains at least two points (the global maximum and global minimum).