I am a student at SMU and I have been working on this problem for a couple of hours and can't seem to figure it out. This question has a few parts to it, but the later parts rely on the first one, so maybe if I can get some help on that I will be able to do the others. Here is the question:

Show, using Rolle's Theorm, that for every subinterval of data set there exists some such that

I can see why this would be the case from the MVT, but I'm having a difficult time formulating a proof. Help would be greatly appreciated, then *hopefully* I will be able to get somewhere with the next part to this problem.