Exercise 12, § 2

Let $K$ be a maximal ordered field, let $f$ be a polynomial over $K$ and let $a$ and $b$ be two roots of $F$ such that $a < b$ and such that $F$ has no roots between $a$ and $b$. Show that if $g$ is a rational function over $K$ whose denominator does not vanish for $a \leq x \leq b$ then the equation $f(x)g(x) + f'(x) = 0$ has an odd number of solutions in $(a, b)$ (where each solution is counted with its multiplicity; use Prop. 5 of VI, p. 25). Deduce that if $h$ is a rational function over $K$ having $a$ and $b$ as roots and whose denominator does not vanish in $]a, b[$, then the equation $h'(x) = 0$ has at least one root in $]a, b[$.