Problem Statement: Use the Mean Value Theorem to show that for all .
Proof: Let . We wish to show that for every . First note that for all . Let and consider the interval . Since is continuous and differentiable on we may apply the Mean Value Theorem. According to the MVT there exists a point such that . We know that so . Since we know the previous statement implies that . Since was an arbitrary number greater than we may conclude that for every .
Reflection: This one actually took me a while. My first thought was to use the function as defined above, but instead of looking at the interval I looked at an arbitrary interval . This allowed me to conclude that but that didn’t give me what I wanted. It was after realizing that and that I could use as an endpoint of the interval the proof kind of fell out.