Q:

Suppose $p$ is a polynomial with real coefficients. Then which of the following statements are necessarily true?

Suppose $p$ is a polynomial with real coefficients. Then which of the following statements are necessarily true?

There is no root of the derivative $p$' between two real roots of the polynomial $p$.
There is exactly one root of the derivative $p$' between any two real roots of the polynomial $p$.
There is exactly one root of the derivative $p$' between any two consecutive real roots of the polynomial $p$.
There is at least one root of the derivative $p$' between any two consecutive roots of $p$

I have taken $p(x)=x^2-1$ then $p'(x)=2x$, here 0 is the root of $p'$ that is lying between the roots of $p$ that is -1 and 1. Hence option 1 is wrong.
For option 2, i have  chosen $p(x)=x$. I guess that option 3 is true but i am not able to prove. Also i am not able to give example for option 4. Please help me! 

A:

Hint: Rolle's theortem
If a real-valued function $f$ is continuous on a proper closed interval $[a, b]$, differentiable on the open interval (a, b), and $f(a) = f(b)(=0)$, then there exists at least one c in the open interval $(a, b)$ such that
$f'(c) = 0$.
Now, what about a polynomial and two consequtive roots of it? (The first three statements can be falsified by simple examples.)

