Mean Value Theorem requires f to be continuous on [a,b] and differentiable on (a,b). What does it guarantee?

Mean Value Theorem requires f to be continuous on [a,b] and differentiable on (a,b). What does it guarantee?

• A. There exists c in (a,b) with f'(c) = 0
• B. There exists c in (a,b) such that f'(c) = (f(b) − f(a))/(b − a)
• C. There exists c in [a,b] with f'(c) = (f(b) − f(a))/(b − a)
• D. There exists c in (a,b) with f''(c) = (f(b) − f(a))/(b − a)

Answer: (B)

Mean Value Theorem says that if a function is continuous on [a,b] and differentiable on (a,b), there is a point c in the interior (a,b) where the tangent slope matches the overall average slope over the interval. In symbols, f'(c) = [f(b) − f(a)] / (b − a). This means the instantaneous rate of change at some interior point equals the average rate of change from a to b, so the tangent line at that point is parallel to the secant line joining (a, f(a)) and (b, f(b)). This is the guaranteed result because continuity ensures no jumps at the endpoints and differentiability inside guarantees a well-defined tangent slope everywhere in (a,b). The other ideas aren’t guaranteed: the derivative needing to be zero isn’t implied by MVT (for example, f(x) = x has derivative 1 everywhere). The interior point is essential—the endpoint values aren’t guaranteed to yield the same slope. And nothing in MVT asserts anything about the second derivative.

Mean Value Theorem says that if a function is continuous on [a,b] and differentiable on (a,b), there is a point c in the interior (a,b) where the tangent slope matches the overall average slope over the interval. In symbols, f'(c) = [f(b) − f(a)] / (b − a). This means the instantaneous rate of change at some interior point equals the average rate of change from a to b, so the tangent line at that point is parallel to the secant line joining (a, f(a)) and (b, f(b)).

This is the guaranteed result because continuity ensures no jumps at the endpoints and differentiability inside guarantees a well-defined tangent slope everywhere in (a,b). The other ideas aren’t guaranteed: the derivative needing to be zero isn’t implied by MVT (for example, f(x) = x has derivative 1 everywhere). The interior point is essential—the endpoint values aren’t guaranteed to yield the same slope. And nothing in MVT asserts anything about the second derivative.