Understanding Rolle's Theorem Rolle's theorem applies to a function defined on a closed interval [a, b] that is continuous and differentiable. If the values of the function at endpoints 'a' and 'b' are equal, there exists at least one point c in (a, b) where the derivative equals zero. This concept ensures clarity about conditions like continuity and differentiability for polynomial functions.
Verification Process of Rolle’s Theorem To verify Rolle’s theorem for f(x)=x² over [-1, 1], check if it satisfies all three conditions: continuity on [−1, 1], differentiability on (−1, 1), and equality f(−1)=f(1). Since these hold true with derivatives leading to critical points satisfying f'(c)=0 within this range—verification confirms its validity.
Application Insights into Polynomial Functions Polynomial functions inherently meet criteria such as being continuous everywhere and infinitely differentiable. Applying Rolle's theorem demonstrates how specific intervals yield points where slopes become zero due to symmetry or other properties inherent in polynomials. These insights reinforce understanding mathematical proofs through practical examples.