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 for all $x \in \mathbb{R}$ and all $n \in \mathbb{N}$, we have $$f(x) = \int_{x_{0}}^{x} \frac{(x-t)^{n}}{n!} f^{(n+1)}(t)\, dt$$
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 for all $x \in \mathbb{R}$ and all $n \in \mathbb{N}$, we have
$$f(x) = \int_{x_{0}}^{x} \frac{(x-t)^{n}}{n!} f^{(n+1)}(t)\, dt$$