Problem Statement: Suppose is a real-valued differentiable function such that for every . Show there is at most one such that .
Proof: Suppose to the contrary that there are two such values such that and . Wlog assume that .
Since is differentiable on we may apply the Mean Value Theorem on the interval . MVT implies that there is some point such that
Since we get the above inequality. But this is a contradiction since we are given that for every . Thus, there is at most one point such that .
Reflection: I’m really glad I got this one. When I first saw it I wasn’t exactly sure what was going to happen, but I knew that using MVT would probably be a good idea. Then it all kind of fell out. I’m getting to the point where I know which tools will probably be best to use, even if I don’t necessarily know how the proof is going to play out. This is good I hope!