$f: [0,1] \rightarrow \Bbb R$ be a differentiable non-constant funtion such that $f(0) = f(1)$. Show $\exists$ a point $x \in [0, 1]$ such that $f'(x)$ is rational.
Here's what I've done:
I believe that this fact holds true as a consequence of the Mean Value Theroem and Rolle's Theorem.
As f is differentiable $\implies$ f is continuous over this compact set. This means that f attains a max and a min. Now if either maximum or minimum occurs at a point c in the interior $[a, b]$, then $f'(c)=0$ (by the interior extremum theroem) which is rational.
If I could have some guidance whether I'm on the right track that would be great.
By Rolle, there is $x \in [0,1]$ such that $f'(x)=0.$ Then $f'(x)$ is rational.