Based off of this information, what conclusions can be made about Rolle's Theorem?This contradicts Rolle's Theorem, since f is differentiable, f(−1) = f(1), and f '(c) = 0 exists, but c is not in (−1, 1).This does not contradict Rolle's Theorem, since f '(0) = 0, and 0 is in the interval (−1, 1). This contradicts Rolle's Theorem, since f(−1) = f(1), there should exist a number c in (−1, 1) such that f '(c) = 0.This does not contradict Rolle's Theorem, since f '(0) does not exist, and so f is not differentiable on (−1, 1).Nothing can be concluded.
Question
Based off of this information, what conclusions can be made about Rolle's Theorem?This contradicts Rolle's Theorem, since f is differentiable, f(−1) = f(1), and f '(c) = 0 exists, but c is not in (−1, 1).This does not contradict Rolle's Theorem, since f '(0) = 0, and 0 is in the interval (−1, 1). This contradicts Rolle's Theorem, since f(−1) = f(1), there should exist a number c in (−1, 1) such that f '(c) = 0.This does not contradict Rolle's Theorem, since f '(0) does not exist, and so f is not differentiable on (−1, 1).Nothing can be concluded.
Solution
Based on the information provided, we can conclude that Rolle's Theorem states that 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 interval (a, b) such that the derivative f'(c) = 0.
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The first statement contradicts Rolle's Theorem because even though f is differentiable and f(-1) = f(1), the value of c where f'(c) = 0 is not within the interval (-1, 1).
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The second statement does not contradict Rolle's Theorem. The function's derivative at 0 is 0, and 0 is within the interval (-1, 1).
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The third statement contradicts Rolle's Theorem. Even though f(-1) = f(1), there is no c in the interval (-1, 1) where f'(c) = 0.
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The fourth statement does not contradict Rolle's Theorem because the function is not differentiable on the interval (-1, 1), which is a requirement of Rolle's Theorem.
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The last statement is incorrect. We can make conclusions about Rolle's Theorem based on the provided information.
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