What is the significance of Rolle’s theorem in differentiability?
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Rolle's Theorem is significant in understanding the behavior of differentiable functions. It states that if a function is continuous on [a, b], differentiable on (a, b), and f(a)=f(b), then there exists at least one point c in (a, b) where f'(c)=0. This means the function has a horizontal tangent (slope zero) at some point between a and b. Rolle's Theorem helps in proving other important results like the Mean Value Theorem and is useful in analyzing turning points, ensuring the existence of critical points in various real-life applications involving motion or optimization.
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