Final answer:
The correct answer according to the Mean Value Theorem for integrals is option B: f(c) = (f(b) + f(a)) / 2. This formula represents the average value of the function f(x) over the interval [a, b].
Step-by-step explanation:
The question is related to the application of the Mean Value Theorem for integrals to determine a function value at a specific point. The Mean Value Theorem for integrals states that there exists a number c in the interval [a, b], where the function f(c) is equal to the average value of the function over the interval. Calculating this average value involves the integral of f(x) over the interval [a, b], divided by the length of the interval.
Looking at the provided options, we can eliminate options A, C, and D, as they refer to the derivative of the function or incorrect operations involving derivatives. The correct option that applies the Mean Value Theorem for integrals can be stated as:
Option B: f(c) = (f(b) + f(a)) / 2
This option represents the average value of a continuous function f(x) on the interval [a, b]. According to the Mean Value Theorem for integrals, there must be at least one point c in the interval where the value of f(c) is exactly this average. Therefore, the correct answer to the question is option B.