Given a rational function $R(x)=P(x)/Q(x)$ with real coefficients, is it true that there exists an $M>0$ such that, for every $k\geq 0$, the restrictions $R^{(k)}|_{(-\infty,-M]}$ and $R^{(k)}|_{[M,\infty)}$ of the $k$-th derivatives $R^{(k)}(x)$ are all monotonic?Or, in other words, is all the interesting stuff happening inside $[-M,M]$ to all orders?
If we take the $k$-th derivative of $P/Q$, it can be proven by induction that it will be of the form $H/Q^{2^k}$ with $\deg H\leq p+(2^k-1)q-k$ where $p=\deg P$ and $q=\deg Q$. Idk if this is of any help.
(Edit: notice that we're asking if $\exists M\forall k \ldots$ and not just if $\forall k \exists M_k\ldots$; the latter is easy because a polynomial changes sign only finitely many times)