Suppose $E=\{f\in C^1[0,1]$: $f(0)=0$,$f(1)=1\}$. Find the maximal $u\in \mathbb{R}$,such that forall $f\in E$, \begin{align*} \int_0^1|f'(x)-f(x)|dx \geqslant u. \end{align*}
I can prove that $u\geqslant \frac1e$,since \begin{align*} \frac1e=e^{-x}f(x)\Big|_0^1=\int_0^1e^{-x}(f'(x)-f(x))dx\leqslant \int_0^1|f'(x)-f(x)|dx . \end{align*} But I don't know how to find the maximal $u$.