Let be continuous. Prove that there exists such that
I tried several things with this problem. I first tried a new function
Evaluating and didn't really work.
Since I want to find a point where and another where , I started thinking how I can make ,or
Since the LHS is the gradient from the point to , I need to somehow prove that that
is always less than the steepest gradient from to . However, I am stuck here. Any help?