Find the maximum of a functional over all $f$ on $[0,1]$, where $0 \leq f(x) \leq x$ (difficult)

Question

Let $X$ be the space of all continuous functions $f : [0,1] \rightarrow \mathbb{R}$ with $0 \leq f(x) \leq x$ for every $x\in [0,1]$. Consider the functional $$F(f)=\int_{0}^1 f(x)^2 dx - \left( \int_0^1 f (x) dx \right)^2, \quad f\in X.$$ Does $F$ admit a maximum in $X$?