Given $p\in\mathbb{Z}[x]$, assumed to have a real root, let $r$ be the largest real root of $p$. Now, given $f\in\mathbb{Z}[x]$ (without loss, of lesser degree than $p$), I would like to find out the sign of $f(r)$, using exact arithmetic. Are there known algorithms for that?

Fundamental problems in algorithmic algebra. $\endgroup$