Bounding $\int_0^1 f(x) dx$, given $\int_0^1 f'(x)^2 dx \leq 1$ and $f(0) = 0$.

Question

Let $S$ be the set of all differentiable function $f \colon [0,1] \rightarrow \mathbb{R}$ such that $\int_0^1 f'(x)^2 dx \leq 1$ and $f(0) = 0$. Define $J(f) := \int_0^1 f(x) dx$. Show that $J$ is bounded on $S$, find its supremum and see if there is a function $f_0$ in $S$ at which $J$ attains its maximum value.

Answer 1

To get you started:

1. Integration by parts yields $J(f)=\int_0^1 (1-x)f\,'(x)\,dx$.

2. Cauchy-Schwarz will appear sooner or later.