Can a first derivative of a function have more roots than the original function?

Question

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

true?

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).

Answer 1

Take $f(x)=3x^2-3$ and let be $g(x)$ a primitive of $f$, for example $x^3-3x+C$. If $C$ is large enough, $g$ will have only a root and its derivative (namely $f$) has two.

EDIT: With $C>2$, $g$ has only one real root.

Answer 2

$f(x)=5+\sin x$ has no roots, and its derivative has infinitely many.

Answer 3

The answer is affirmative even for polynomial functions, e.g. $y=1+x^2$.