Rolle Theorem Continued

graph TB A[Rolle's Theorem (Continued)] A --> B[Definition] A --> C[Assumptions] A --> D[Proof] B --> E[If a function f is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one c in the open interval (a, b) such that f'(c) = 0.] C --> F[The function f is continuous on the closed interval [a, b]] C --> G[The function f is differentiable on the open interval (a, b)] C --> H[f(a) = f(b)] D --> I[Consider any function that meets the assumptions of Rolle's Theorem] D --> J[By the Mean Value Theorem, there exists at least one c in the open interval (a, b) such that f'(c) = (f(b) - f(a)) / (b - a)] D --> K[Since f(a) = f(b), f'(c) = 0 for at least one c in the open interval (a, b)]