Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a function of class $C^{\infty}$ with compact support. For all $n \in \mathbb{N}$, we set
$$M_{n} = \sup_{x \in \mathbb{R}} \left|f^{(n)}(x)\right| = \left\|f^{(n)}\right\|_{\infty}$$
and we seek properties satisfied by the sequence $(M_{n})$. In this part we assume that $f$ is not identically zero.
Show that there exists $x_{0} \in \operatorname{Supp}(f)$ such that for all integers $n \geqslant 0$, $f^{(n)}(x_{0}) = 0$.