Exercise 16, § 3

Let $f$ be a real function $n$ times differentiable on $I = ]-1,+1[$, and such that $|f(x)| \leq 1$ on this interval.

a) Show that if $m_k(\lambda)$ denotes the minimum of $|f^{(k)}(x)|$ on an interval of length $\lambda$ contained in $I$ then one has

(Note that if the interval of length $\lambda$ is decomposed into three intervals of lengths $\alpha, \beta, \gamma$, one has

b) Deduce from a) that there exists a number $\mu_n$ depending only on the integer $n$ such that if $|f'(0)| \geq \mu_n$, then the derivative $f^{(n)}(x)$ vanishes on at least $n-1$ distinct points of $I$ (show by induction on $k$ that $f^{(k)}$ vanishes at least $k-1$ times on $I$).