**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.

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