f is uni-variate polynomial of degree d.
I am interested in lower bounding modulus first derivative of f (i.e. |f'|)in interval [a,b] given the promise that **in interval [a,b] f' don't have any root.**


Like in **Markoff theorem**..to upper bound first derivative of polynomial in open interval (a,b)
 is     
       |f'(x)|<=(2*M*d*d)/(b-a)
where M is upper bound of f in (a,b),but I don't know proof of this theorem.

My question is analogues to this inequality can we lower bound first derivative of f.



Thanks
Ram