Examples establishing conditions for MVT  Summary and Q&A
TL;DR
The mean value theorem allows us to find a number within an interval where the derivative of a function is equal to the average rate of change.
Questions & Answers
Q: What is the mean value theorem?
The mean value theorem states that if a function is continuous over a closed interval and differentiable over an open interval, then there must be a number within the closed interval where the derivative of the function is equal to the average rate of change.
Q: What conditions need to be met to apply the mean value theorem?
The function must be continuous over the closed interval and differentiable over the open interval.
Q: Why is it important to meet the conditions of the mean value theorem?
Meeting the conditions ensures that the conclusion drawn from applying the mean value theorem is valid. Without meeting the conditions, the theorem cannot guarantee the existence of a number where the derivative is equal to the average rate of change.
Q: Which choice satisfies the conditions for applying the mean value theorem in this case?
The choice that states "g is differentiable over the open interval from 1 to 1" satisfies the conditions of the theorem, as it covers the open interval [0, 1] that we are interested in.
Summary & Key Takeaways

Rafael claims that there must be a number within the interval [0, 1] where the derivative of the function g is equal to 5, based on the average rate of change between x = 0 and x = 1.

To apply the mean value theorem, we need to meet the conditions of the theorem, which include the function being continuous over the closed interval and differentiable over the open interval.

Of the given choices, the condition that makes Rafael's claim true is that g is differentiable over the open interval from 1 to 1.