TITLE: Do inequalities hold under differentiation?
QUESTION [21 upvotes]: If I have the inequality $f \leq g$, does this imply $f'\leq g'$?
The reason I am asking is because I am using this fact to prove a question with Taylor theorem.

REPLY [9 votes]: Take two functions whose derivative have whatever inequality you wish (ie., $f^\prime > g^\prime$ or vice versa). Then just add or subtract a constant to one of the functions to make it larger or smaller than the other.
In short, you can very easily create pairs of functions whose values obey one inequality relation and their derivatives hold another.  Or the same.