Boundary Values & Fourier Series · Superchallenge problem

Find the largest constant $C$ such that $\displaystyle\int_0^2 u'(x)^2\,dx \ge C\int_0^2 u(x)^2\,dx$ for every smooth function $u$ on $[0,2]$ with $u(0) = 0$ (no condition at $x = 2$).