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 $$M_{n} = \sup_{x \in \mathbb{R}} \left|f^{(n)}(x)\right| = \left\|f^{(n)}\right\|_{\infty}$$ In this part we assume that $f$ is not identically zero, and $x_{0} \in \operatorname{Supp}(f)$ is such that for all integers $n \geqslant 0$, $f^{(n)}(x_{0}) = 0$. Show that if there exist constants $A > 0$ and $B > 0$, and a subsequence $(n_{j})_{j \geqslant 1}$ such that $M_{n_{j}} \leqslant A B^{n_{j}} (n_{j})!$, then $f$ is identically zero on the interval $]x_{0} - 1/B,\, x_{0} + 1/B[$.
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
$$M_{n} = \sup_{x \in \mathbb{R}} \left|f^{(n)}(x)\right| = \left\|f^{(n)}\right\|_{\infty}$$
In this part we assume that $f$ is not identically zero, and $x_{0} \in \operatorname{Supp}(f)$ is such that for all integers $n \geqslant 0$, $f^{(n)}(x_{0}) = 0$.
Show that if there exist constants $A > 0$ and $B > 0$, and a subsequence $(n_{j})_{j \geqslant 1}$ such that $M_{n_{j}} \leqslant A B^{n_{j}} (n_{j})!$, then $f$ is identically zero on the interval $]x_{0} - 1/B,\, x_{0} + 1/B[$.