We fix two distinct real numbers $x_1 < x_2$ in $[0,1]$. For all $x \in [0,1]$ and $f \in \mathcal{C}^2([0,1])$, prove the inequality
$$\left|f^\prime(x) - \frac{f\left(x_2\right) - f\left(x_1\right)}{x_2 - x_1}\right| \leqslant \left\|f^{\prime\prime}\right\|_\infty.$$