Let $E$ be the set of all functions $u : [0, 1] \to R$ such that $u(0) = 0$ and $u$ satisfies a Lipschitz condition with Lipschitz constant $1$.
Define φ : E → R according to the formula:
$$ \phi(u) = \int_0^1 (u(x)^2 - u(x)) dx $$ Prove that there exists a function $u \in E$ at which $\phi(u)$ attains an absolute maximum
Here's what I know:
The Lipschitz condition implies that the functions $u$ are equicontinuous
Arzela-Ascoli implies that the set $E$ has a convergent subsequence.
The fact $u(0)=0$ screams out to me that I need to use the Fundamental Thm of Calculus
But other than that, I'm really unsure how to begin.