Rolle Theorem Continued
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)]
