Bernstein’s theorem

Let $f \in \mathcal{C}^{\infty}(]-a,a[,\mathbb{R})$ be such that $\forall n \in \mathbb{N}, f^{(n)} \geqslant 0$. The goal is to show that $f$ is analytic, that is, expandable as a power series at every point of its domain of definition.

1. By considering $g_{x_0} : x \longmapsto f(x+x_0)$ with $x_0 \in \, ]-a,a[$, so that $g_{x_0}$ is defined on $]-a+|x_0|, a-|x_0|[$, show that it suffices to show that $f$ is expandable as a power series in a neighborhood of $0$ in order to conclude.

We now seek to show that $f$ is expandable as a power series in a neighborhood of $0$.

2. Show that:
   $$\forall n \in \mathbb{N}, \forall x \in \, ]-a,a[, \quad f(x) = \sum_{k=0}^{n} \frac{f^{(k)}(0)}{k!} x^k + R_n(x)$$where $R_n$ will be expressed by means of an integral.

3. Consider $b \in \, ]0,a[$. Show that, for every natural number $n$ and every real number $x$ of $]-b,b[$, we have the bound:
   $$\forall n \in \mathbb{N}, \forall x \in \, ]-b,b[, \quad |R_n(x)| \leqslant \left(\frac{|x|}{b}\right)^{n+1} R_n(b) \leqslant \left(\frac{|x|}{b}\right)^{n+1} f(b)$$

Then deduce Bernstein’s theorem.

4. Application: Let $f : [0,1] \to \mathbb{R}$ be not identically zero such that $\forall n \in \mathbb{N}, f^{(n)} \geqslant 0$. Show that $f$ vanishes at most once.