Let $a,b \in \mathbb{R}$ be such that $a\lt b $. Suppose that $f:[a,b]→\mathbb{R}$ is a continuous function on $[a,b]$ and is differentiable on $(a,b)$ and that
$f'(x) \gt 1$ , for all $x\in(a,b)$.
Use Rolle's Theorem to show that $f$ has at most one fixed point in (a,b), that is a point $x_o\in(a,b)$ such that $f(x_0)=x_0$.
Hint: You may find it helpful to argue using proof by contradiction.
My solution so far:
Rolle's Theorem states that if $f$ is a continuous function on $[a,b]$ and is differentiable on $(a,b)$ and that $f(a)=f(b)$, then there is a point $c\in(a,b)$ such that $f'(c)=0$.
Assume for a contradiction, there are two fixed points in $(a,b)$, that is, $x_0,x_1\in(a,b)$ such that $f(x_0)=x_0$ and $f(x_1)=x_1$.
Now, I don't know how to proceed from this because $f(x_0)\ne f(x_1)$, so surely you can't apply Rolle's Theorem here.
Note: This is not a duplicate question as this question explicitly asks to use Rolle's Theorem.