Let $f: I \longrightarrow \mathbb{R}$ differentiable with $I$ an interval. Prove that between two consecutive roots of $f'$, there exists at most one root of $f$.
$\textbf{Solution:}$ Let $x_{1} < x_{2}$ be such that $f'(x_{1}) = f'(x_{2}) = 0$ and $x_{1}, x_{2}$ be consecutive roots. If there exist $c_{1} < c_{2} \in (x_{1}, x_{2})$ such that $f(c_{1}) = f(c_{2}) = 0$, then, by the Rolle's Theorem, there exists $c \in (c_{1}, c_{2})$ where $f'(c) = 0$, an absurdity.
Is the idea correct?