Given an ordered field, we can view it as a field formally equipped with some analytical notions which come from $\mathbb R$, like order and derivative. So I'm curious if $F$ also carries some analytical properties of $\mathbb R$ such as several mean value properties. To begin with, I'd like to know:
Let $(F,+,×,\leqslant)$ be an ordered field, $a,b,c\in F$, $f\in F[x]$, can we prove that:
If for any $c\in(a,b)$ we have $f'(c)\geqslant0$ (resp. $f'(c)>0$), then $f(a)\leqslant f(b)$ (resp. $f(a)<f(b)$).