**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!