Exercise 13, § 2

Let $K$ be a maximal ordered field, let $h$ be a rational function over $K$ and let $[a, b]$ be a closed interval in which $h$ is defined. Show that there exists $c \in ]a, b[$ such that $h(b) - h(a) = (b - a)h'(c)$ (use Ex. 12). Deduce that $h$ is an increasing function in $[a, b]$ if and only if $h'(x) \geq 0$ in this interval (to see that this condition is necessary, split the interval by the roots of $h'(x) = 0$).