Question

How to determine if a function satisfies the Mean Value Theorem?

Answer 1

To determine if a function satisfies the Mean Value Theorem, the function must be continuous on the closed interval [a, b], differentiable on the open interval (a, b), and have a point c in (a, b) where f'(c) equals the average rate of change over [a, b].

Step-by-step explanation:

To determine if a function satisfies the Mean Value Theorem (MVT), you should follow these steps:

1. Ensure the function is continuous on the closed interval [a, b].
2. Check that the function is differentiable on the open interval (a, b).
3. Once the first two conditions are met, MVT states there exists at least one c in the interval (a, b) such that the function's derivative at c equals the average rate of change of the function over [a, b]. Mathematically, f'(c) = (f(b) - f(a)) / (b - a).

If these conditions are satisfied, the function meets the criteria of the MVT. This theorem is a standard result in differential calculus used to predict the existence of certain points given the global behavior over an interval.