Non existence of a non vanishing 1-form on a compact manifold without boundary

Question

Let $M$ be a compact manifold without a boundary and let us assume $\omega$ is a non-vanishing 1-form on $M$ that is exact. Then there is a 0-form, or a function to be precise, $f:M\to \mathbb{R}$ such that $df=\omega\neq0$. I am trying to get a contradiction but I cannot seem to do it. I guess if $f$ has any critical points then $df=0$ at such points but does $f$ have critical points?

Answer 1

As $M$ is compact $f$ takes a maximum somewhere, and there $df=0$.