Final answer:
The statement is false; the average value of a function is not necessarily attained at a point within the interval [a, b].
Step-by-step explanation:
The statement that the average value of a function is always attained at a specific point within the interval [a, b] is false. This is a common misconception based on the Mean Value Theorem, which applies to continuous differentiable functions and guarantees that there is at least one point c in the open interval (a, b) where the instantaneous rate of change (derivative) is the same as the average rate of change on the interval. However, it does not guarantee that the value of the function at c will be the same as the average value of the function over the interval [a, b]. Consider, for instance, a function that represents the speed of a car on a trip. The average speed of the car over the entire trip doesn't have to be the speed of the car at any specific point in time.