A continuously differentiable function $f:[0,1]\to[0,1]$ has the properties
(a) $f(0)=f(1)=0$
(b) $f'(x)$ is a non-increasing function of $x$.
Prove that the arc-length of the graph does not exceed $3$.
As I understand from the question that we want to prove that $\int_{0}^{1}f(x)dx <3$.
The first property give the conditions of Rolle's theorem implies that $f'(c)=0$, $c \in(0,1)$.
The second property gives the clue of the max value exists at $c$.
I tried to use first mean value theorem of integral, but no conclusion find.
Is there is any other technique to solve this question?