Let be a continous function and . If be differentiable over , prove that there exist such that .
First, let . Apply MVT to this function on : where c between 0 and 1.
Clearly, and . Also, by the FTC, . Hence, we have: .
Let's now apply the MVT for the function on the interval (or the other way around). We get: .
Therefore, i.e. .
That what I was able to reach so far...Inspire from this. The continuation must be in a similar manner.
Hope this helps