Real function/Theorem of Rolle/Fact

Theorem of Rolle

Let ${}a<b$, and let

$f\colon [a,b]\longrightarrow \mathbb {R}$

be a continuous function, which is differentiable on ${}]a,b[$, and such that ${}f(a)=f(b)$.

Then there exists some

${}c\in {]a,b[}$, such that

${}f'(c)=0\,.$

Proof, Write another proof