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.
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.