Problem Statement: Let be a twice differentiable real valued function defined on . Suppose that with and . Prove there exists a such that .
Proof: Since is twice differentiable on we know that is differentiable on , which implies that is continuous on .
Since is differentiable on and we may apply the Mean Value Theorem on those intervals. The MVT implies that there exists and such that
Note that since it follows that . Since it follows that . Similarly we see that and since it follows that . As stated above is continuous on and since it is continuous on . So, by the Intermediate Value Theorem it follows that there is some point such that . Now apply the MVT on the interval . Then there exists some such that
As stated above so . By construction and so . Thus .
Reflection: When I first approached this problem I knew I had to use the MVT. I got stuck though after finding . I tried to use MVT again on but I couldn’t guarantee that the difference in the numerator was positive. The key for this problem rested in the fact that differentiable implies continuous. Since we had a continuous derivative function and we knew it was positive and negative at two points, we were guaranteed there was a point in between where the derivative was zero. After there it all fell out nicely.