We fix two distinct real numbers $x_1 < x_2$ in $[0,1]$. Deduce that, for any function $f \in \mathcal{C}^{2}([0,1])$, we have
$$\left\|f^{\prime}\right\|_{\infty} \leqslant \left\|f^{\prime\prime}\right\|_{\infty} + \frac{\left|f\left(x_{1}\right)\right| + \left|f\left(x_{2}\right)\right|}{x_{2} - x_{1}}.$$