Show that, for every function $f:[0,1] \rightarrow \mathbb{R}$ of class $\mathcal{C}^1$ such that $f(0) = 0$, we have $$\forall x \in [0,1] \quad |f(x)| \leqslant \sqrt{x \int_0^x (f'(t))^2\,\mathrm{d}t}$$
Show that, for every function $f:[0,1] \rightarrow \mathbb{R}$ of class $\mathcal{C}^1$ such that $f(0) = 0$, we have
$$\forall x \in [0,1] \quad |f(x)| \leqslant \sqrt{x \int_0^x (f'(t))^2\,\mathrm{d}t}$$