Assume $f(x)$, with $x \in [a,b]$. Take $u$ so that $f(a)<u<f(b)$. By the Intermediate value theorem, we know that $f(x)$ crosses $u$ at least once. My question is, given some extra information about $f^{(n)}(x)$ (derivative of $f$), can I predict a higher-bound of the number of times the function crosses $u$?
For example, if I tell you that $f'(x)>0$ over all its support, then we can assure that the number of crossing points is just one (does this has a name? because the IVT only states that it is at least one but not just one since it doesn't rely on the extra slope information). But if I tell you that:
- $f'(x)>0$ for $x \in [a,w)$
- $f'(x)=0$ for $x=w$
- $f'(x)<0$ for $x \in (w,z)$
- $f'(x)=0$ for $x=z$
- $f'(x)>0$ for $x \in (z,b]$
then we know that the maximum number of crossing points its 3 (when $w<u<z$).
Is there a theorem that states this?