The Wronskian & Variation of Parameters · Superchallenge problem

Let $f$ be continuous on $[0,1]$ with $\displaystyle\int_0^1 f(t)^2\,dt=1$, and let $y$ solve $y''=f$ with $y(0)=0$, $y(1)=0$. What is the largest possible value of $y\!\left(\dfrac12\right)$?