This is a general question. Function is to be considered differentiable on some domain.
More specifically, I am given a function $f(x)$ which is twice differentiable and has three distinct real roots. Is the statement
$f '(x)$ has at most three real distinct roots
EDIT: I'm sorry, I meant to ask a question for only those functions that do have roots. For example, $f(x)$ has $n$ roots, can $f'(x)$ have more than $n$ roots (again function is to be considered differentiable and roots are real and all distinct).