Let $f:\mathbb{R}\to \mathbb{R}$ be a $n$ times continuously differentiable function such that $f$ and $f^{(n)}$ are bounded. Show that there is a constant $C$ such that $$\big\|\,f^{\left(i\right)}\big\|_\infty=C\left(\big\|\;f\,\big\|_\infty+\big\|\,f^{\left(n\right)}\big\|_\infty\right)$$ for $1\leq i\leq n-1$, where $\displaystyle\,\left\|\;f\,\right\|_\infty:=\sup_{x\in\mathbb{R}} \left\lvert\, f\left(x\right)\right\rvert$ is the sup-norm.
I tried to use Taylor series with remainder, but cannot solve it. It seems that we need a theorem of Markoff which can be found here. I cannot find the original paper in which Markoff proved the theorem.
