Answer: Continuous on [a,b], differentiable on (a,b), and f(a) = f(b).
- A Differentiable, with continuity treated as not strictly required
- B Continuous on [a,b], differentiable on (a,b), and f(a) = f(b)
- C Equal at the endpoints mainly, with continuity or differentiability not required
- D Continuous on [a,b] mainly, without any differentiability condition
Correct answer: B. Continuous on [a,b], differentiable on (a,b), and f(a) = f(b)
Explanation: Rolle's theorem: if f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then there exists c in (a,b) with f'(c) = 0.
As point Q slides along the curve toward P, the secant line PQ rotates into the tangent line at P, whose slope is the derivative.
Concept context
Rate of change, differentiation rules, and applications