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I think you need to be a bit more precise. For example, choose any polynomial $g(x)$ that does not vanish on $[a,b]$, let $f'(x)=(x-a+\epsilon)g(x)$, let $F(x)$ be an antiderivative of $f'(x)$, and let $f(x)=F(x)+C$ for some constant $C$. Choosing $\epsilon$ small, you can make $f'(a)=\epsilon g(a)$ small, while choosing an appropriate $C$, you can make $\sup |f(x)|$ or $\inf |f(x)|$ on the interval $[a,b]$ to be anything that you want. So it's really not clear what sort of lower bound you have in mind.