Rolle's Theorem and Mean Value Theorem

Rolle's Theorem

• For a function f continuous on [a, b] and differentiable on (a, b), with f(a) = f(b), there exists a point c in (a, b) such that f'(c) = 0.

Mean Value Theorem

• Extends Rolle's Theorem by considering the average change of a function across an interval where the endpoints are not necessarily equal.
• Guarantees a point where the derivative matches the average change.