Does there exists a only one time differentiable function $f$ in an interval $[a,b]$ whose derivative is monotone increasing in $[a,b]$ ? That is, does there exists a function $f$ in $[a,b]$ such that $f'$ exists for all $x\in [a,b]$ and $f'$ is monotone increasing in $[a,b]$ but $f''$ does not exists at some point in $[a,b]$.
I guess there exists such a function. I'm trying to define such function but unable!!