Which condition is not required for the Mean Value Theorem?

Which condition is not required for the Mean Value Theorem?

• A. f is continuous on [a,b]
• B. f is differentiable on (a,b)
• C. f(a) = f(b)
• D. There exists c in (a,b) with f'(c) equal to the average rate of change

Answer: (C)

The essential idea is that if a function is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then somewhere inside the interval the tangent slope matches the average slope over the interval. In symbols, there exists c in (a, b) with f'(c) = [f(b) − f(a)] / (b − a). The condition that the endpoint values are equal is not required for this guarantee. If f(a) = f(b), the average rate of change is zero, and the conclusion becomes there is a point where the derivative is zero—this is Rolle's theorem, a special case arising when the endpoints happen to match. So the endpoint-equality condition is not a general requirement for the Mean Value Theorem.

The essential idea is that if a function is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then somewhere inside the interval the tangent slope matches the average slope over the interval. In symbols, there exists c in (a, b) with f'(c) = [f(b) − f(a)] / (b − a). The condition that the endpoint values are equal is not required for this guarantee. If f(a) = f(b), the average rate of change is zero, and the conclusion becomes there is a point where the derivative is zero—this is Rolle's theorem, a special case arising when the endpoints happen to match. So the endpoint-equality condition is not a general requirement for the Mean Value Theorem.