Let $f$ be a real function. Which conditions on $f$ let me give an upper bound to the number of roots of $f$? I'm intentionally vague on the conditions of $f$, everything it's ok. I would like to have some non-trivial general result, I think this statement should be true:
If $f \in C^k(I)$ and $f^{(k)}$ has exactly $m$ roots then $f$ has at most $m+k$ roots.
Anything better?
By the MVT, for any two roots $x,y$ of $f$, there is a root of $f'$ in $(x,y)$. Thus, if $f$ has $n$ roots, $f'$ has at least $n-1$ roots. Inductively, this means $f^{(k)}$ has at least $n-k$ roots. Hence $n>m+k$, would contradict the fact that $f^{(k)}$ has exactly $m$ roots.