Last week i attended a competition with some friends and there was the following problem:
Let $f$ be a function with continuous derivative over $[0,1]$ such that $f(0)=0$ and $f(1)=1$. Prove that $$\int_0^1|f(t)-f'(t)|dx\geq\frac{1}{e}$$
A friend of mine solved it by using the auxiliary function $g(t)=e^{-t}f(t)$, taking its derivative and then integrating it.
When i was solving the problem i remember a course in which we used a variational approach to minimizing this kind of integrals that ended up in with the Lagrangian of the integrand. I tried using this method in the problem but could not work the lagrangian out
¿Could there be an approach to this problem using the calculus of variations?
Thanks in advance.