Problem Statement: Let be differentiable. Let such that and . If is between and then there exists such that .
Proof: Wlog assume . Fix . Define . Then . Note that and since .
and so for sufficiently close to , for . Similarly, and so for sufficiently close to , for .
Since is continuous on , which is a compact set, we know that attains both it’s max and min on . We know from above that does not attain it’s max when or . So, let such that for every . Then is the maximum, thus, . Thus, and so it follows that .
Reflection: This one was not straightforward at all. I think that after doing this problem I would see the trick, but right away I didn’t see it. When I tried to attack it directly I didn’t make any real progress on the proof. So, maybe this is a good technique for me to have up my sleeve; when trying to prove something about the derivative create a function you know information about.