In this part, $E_1$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, of class $\mathcal{C}^1$ piecewise, and satisfying $f(0) = f(1) = 0$. We denote by $N$ the norm associated with the inner product $(f \mid g) = \int_0^1 f'(t) g'(t) \, \mathrm{d}t$. Show that, for every function $f : [0,1] \rightarrow \mathbb{R}$ of class $\mathcal{C}^1$ such that $f(0) = 0$, we have $$\forall x \in [0,1] \quad |f(x)| \leqslant \sqrt{x \int_0^x (f'(t))^2 \, \mathrm{d}t}$$
In this part, $E_1$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, of class $\mathcal{C}^1$ piecewise, and satisfying $f(0) = f(1) = 0$. We denote by $N$ the norm associated with the inner product $(f \mid g) = \int_0^1 f'(t) g'(t) \, \mathrm{d}t$.\\
Show that, for every function $f : [0,1] \rightarrow \mathbb{R}$ of class $\mathcal{C}^1$ such that $f(0) = 0$, we have
$$\forall x \in [0,1] \quad |f(x)| \leqslant \sqrt{x \int_0^x (f'(t))^2 \, \mathrm{d}t}$$