Integral inequality (Cauchy-Schwarz)

Let $u\in \mathcal{C}^1[a,b]$ be such that $u(a)=u(b)=0$. Show that

$$\int_a^b u^2(x)dx\leq (b-a)^2\int_a^b (u')^2(x)dx$$

using the Schwarz’s inequality.

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