Given two statements:

- The integral
because

- The integral
because, by the weighted mean value theorem (Theorem 3.16 in Apostol), there exists a in such that

Identify which of the two statements is false, and explain why.

Statement (b) is false since we may not apply the weighted mean value theorem in this case. The weighted mean value theorem requires the function to not change sign on the interval . Since does change sign on the interval, we cannot apply the theorem.