The options to this question are $(5,6) ; (0,1) ; (1,2)$ and $(2,5)$.
My attempt :
I assumed the polynomial function to be f'(x) and found f(x) by integrating it. Now, if I am able to find two roots of the function f(x) then according to Rolle's Theorem, there would be one point between those roots where f'(x) would be zero. However, I cannot think of a way to find the constant of integration and precisely determine the roots. I am also not able to find any way of using other mean value theorems in these type of questions
Please note that I am looking for a procedure which only uses Mean Value theorems (Rolle's Theorem, Lagrange's Theorem etc) and basic concepts of polynomials.
All help is appreciated.
Hint: If $f(a)\cdot f(b)<0$ then $f$ has a root in $(a,b)$ if $f$ continuous.