When $[a,b]=[-1,1]$, the $\inf$ on the right-hand side is attained (uniquely) by the monic Chebyshev polynomial $T_{n+1}$. It is well known that its roots belong to $[-1,1]$ and are simple. 

For a general interval $[a,b]$, using the linear map 
$$f(x)=\left(\frac{2}{b-a}\right)x-\left(\frac{b+a}{b-a}\right)$$
that sends $[a,b]$ onto $[-1,1]$, it is clear that the $\inf$ on the right-hand side is attained by
$$
\left(\frac{b-a}{2}\right)^{n+1} T_{n+1}(f(x)),$$
which, of course, has all its (simple) roots in $[a,b]$.