Real function/Theorem of Rolle/Fact/Proof

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