Let $f\in C^{(n)}((-1,1))$ and $\sup_{-1<x<1}|f(x)|\le 1.$ Let $m_k(I)=\inf_{x\in I}|f^{(k)}(x)|$ where $I$ is an interval contained in $(-1,1)$. Show that:
a) if $I$ is partitioned into three successive intervals $I_1,I_2,I_3$ and $\mu$ is the length of $I_2$ , then $m_k(I)\le \frac 1\mu (m_{k-1}(I_1)+m_{k-1}(I_3))$
b) if $I$ has length $\lambda$, then $$m_k(I)\le \frac{2^{\frac{k(k+1)}{2}}k^k}{\lambda^k}$$
c) there exists a number $\alpha_n$ depending only on $n$ such that if $|f'(0)|\ge \alpha_n$, then the equation $f^{(n)}(x)=0$ has at least $n-1$ distinct roots in $(-1,1)$
I have managed to prove a) and found proof of part b) (here: Difficulty in proving this inequality)but I haven't managed to find an answer for part c). So I would like an answer for part c). Any help would be greatly appreciated.
(Although c)'s answer couldn't be found here but it would give you a better idea of it here An exercise from Zorich's book (edit and add a) and b)) )