Let , $f:[0,4]\to [1,3]$ be a differentiable function such that $f'(x)\not=1$ for all $x\in [0,4]$. Then which is correct ?
(A) $f$ has at most one fixed point.
(B) $f$ has unique fixed point.
(C) $f$ has more than one fixed point.
Here, $f:[0,4]\to [1,3]\subset [0,4]$ is continuous and $[0,4]$ is compact convex. So by Brouwer's fixed point theorem $f$ has a fixed point. But I am unable to use the condition $f'(x)\not=1$ and how I can conclude that how many fixed points are there for $f$ ?