What is Roll's theorem?
Understand the Problem
The question is asking about Roll's theorem, which is a key concept in calculus relating to the behavior of differentiable functions. It typically requires an explanation of the theorem's statement and its implications.
Answer
Rolle's theorem states that if a function f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), there exists c in (a, b) where f'(c) = 0.
Rolle's theorem states that if a function f is continuous on the closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there is a point c in (a, b) such that f'(c) = 0.
Answer for screen readers
Rolle's theorem states that if a function f is continuous on the closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there is a point c in (a, b) such that f'(c) = 0.
More Information
Rolle's theorem is a special case of the Mean Value Theorem, which is a critical calculus concept used to understand the behavior of differentiable functions over an interval.
Tips
A common mistake is failing to check that the function meets all the hypothesis conditions: continuous on the closed interval and differentiable on the open interval.
Sources
- Rolle's theorem - Wikipedia - en.wikipedia.org
- Rolle's theorem | Definition, Equation, & Facts - Britannica - britannica.com
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