TITLE: Polynomial inequalities in ordered fields
QUESTION [6 upvotes]: Let $p(x)$ be a polynomial over an ordered field. If $p'(x)\ge 0$ for all $x$ in an interval over that field, is it true that $p(x)$ is increasing over that interval?

REPLY [3 votes]: It is remarked in A Course in Model Theory: An Introduction to Contemporary Mathematical Logic by Bruno Poizat, page 101 (Google Books link), that the assertion is true for all ordered fields, but that its proof is difficult. He gives the proof for real closed fields just prior to that remark.