Let $C \in ]0,1[$. Using a simple example of a function $f$, show that the interpolation inequality
$$\forall f \in \mathcal{C}^{1}([0,1]), \quad \|f\|_{\infty} \leqslant \left\|f^{\prime}\right\|_{\infty} + C\left|f\left(x_{1}\right)\right|$$
is false.