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
$$\left\|f^\prime\right\|_\infty \leqslant \left\|f^{\prime\prime}\right\|_\infty + \frac{\left|f\left(x_1\right)\right| + \left|f\left(x_2\right)\right|}{x_2 - x_1}.$$