The number of roots of $f'$

Question

$f$ is a polynomial. And has root on $a$ and $b$. There is no roots between $(a,b)$. How do I prove that $f'$ has odd number roots between $(a,b)$?

I can imagine the graph so I can guess there are odd numbers. But how could I prove it ?

Answer 1

This claim need not be true.

You can verify yourself : $f(x)=6x^4-8x^3-12x^2+24x-96$ Between the two roots of the function, its derivative has two roots.