Which statement is true about Rolle's Theorem?

Which statement is true about Rolle's Theorem?

• A. It requires f continuous on [a,b] and differentiable on (a,b) and guarantees a point c with f'(c) > 0
• B. It guarantees a point c with f'(c) = 0 under the stated hypotheses
• C. It applies to any function with a derivative on (a,b)
• D. It requires f(a) ≠ f(b) for the conclusion in (a,b)

Answer: (B)

Rolle's Theorem says that if a function is continuous on the closed interval [a,b], differentiable on the open interval (a,b), and the endpoint values are the same (f(a)=f(b)), then there is at least one point c in (a,b) where the derivative is zero. The reason this happens is that the overall change in function value from a to b is zero, so the average rate of change over [a,b] is zero. By the Mean Value Theorem, there must be some interior point where the instantaneous rate of change matches the average rate, which is 0, giving f'(c)=0. This guarantees a horizontal tangent somewhere between a and b. So the statement that matches this idea is that there exists a point inside where the derivative equals zero. It’s not about the derivative being positive, and it requires equality of the endpoint values, not inequality. Also, having a derivative on (a,b) alone isn’t enough—you need the continuity on [a,b] and f(a)=f(b). For example, f(x) = (x−1/2)^2 on [0,1] is continuous, differentiable, has equal endpoints, and f'(x) = 2x−1, which is zero at x=1/2.

Rolle's Theorem says that if a function is continuous on the closed interval [a,b], differentiable on the open interval (a,b), and the endpoint values are the same (f(a)=f(b)), then there is at least one point c in (a,b) where the derivative is zero. The reason this happens is that the overall change in function value from a to b is zero, so the average rate of change over [a,b] is zero. By the Mean Value Theorem, there must be some interior point where the instantaneous rate of change matches the average rate, which is 0, giving f'(c)=0. This guarantees a horizontal tangent somewhere between a and b.

So the statement that matches this idea is that there exists a point inside where the derivative equals zero. It’s not about the derivative being positive, and it requires equality of the endpoint values, not inequality. Also, having a derivative on (a,b) alone isn’t enough—you need the continuity on [a,b] and f(a)=f(b). For example, f(x) = (x−1/2)^2 on [0,1] is continuous, differentiable, has equal endpoints, and f'(x) = 2x−1, which is zero at x=1/2.