Exercise 19, § 2

Let K be a maximal ordered field and f a nonzero polynomial in K[T]. A necessary and sufficient condition that, for every polynomial $g \in K[T]$, the number of roots of $fg + g'$, counted with multiplicity, not belonging to K, be at most equal to the number of roots of $g$, counted with multiplicity, not belonging to K, is that $f(T) = a - bT$ with $b \geq 0$ in $K$. (To see that the condition is sufficient use Ex. 12; to show that it is necessary, take $g = 1$ and $g = f$).