Suppose that is an open interval and that for all . If , show that the part of the graph of on is never below the tangent line to the graph at .
So, let . Then, is the difference between and the tangent line at . So, what we want to do is show that and . But, and and so if then and if then . From prior comment the conclusion follows.
NOTE! The above is purposefully full of holes! Fill them in!