1) By applying the Mean Value Theorem to f(x)=sqrt x, show that 1/11 < (sqrt 102) -10 < 1/10. This is a sample problem in my course, and how to start this problem is the key thing. The solution says "f is continuous and differentiable for all x>0, so by the mean value theorem, there exists c E (100,102) such that (conclusion of mean value theorem)....etc" Say if you have never seen such a problem before, how can you get the inspiration to pick the interval (100,102) and not something else? How is it even possible to know this ahead of time before you start the proof? Can someone teach me the logic of this choice and most importantly, how you arrived at the choice of picking this particular interval? Thank you very much!