If $f$ is injective on an open interval $(a,b)$, then what properties we can deduce about the zeros of $f'$?
We can not get that $f' \neq 0$. One example is $f(t) = t^3$ and $f'(0)=0$. I think at least we know that $f'$ can not equal $0$ on any subinterval of $(a,b)$. Otherwise, using Rolle's theorem, we can deduce a contradiction.
So, is there some other advanced properties about the zeros of $f'$. For example, I guess would the zeros of $f'$ be isolated?