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
$$\left|f^{\prime}(x) - \frac{f\left(x_{2}\right) - f\left(x_{1}\right)}{x_{2} - x_{1}}\right| \leqslant \left\|f^{\prime\prime}\right\|_{\infty}.$$