Real function/Theorem of Rolle/Fact/Proof

Proof

The statement is true if ${}f$ is constant. So suppose that ${}f$ is not constant. Then there exists some ${}x\in {]a,b[}$, such that ${}f(x)\neq f(a)=f(b)$. Let's say that ${}f(x)$ has a larger value. Due to fact, there exists some ${}c\in [a,b]$, where the function attains its maximum. This point is not on the border. For this ${}c$, we have ${}f'(c)=0$, due to fact.

To fact