Find the value c such that the conclusion of Rolle's Theorem are satisfied for f(x) = 2x² - 2 on [-3, 3].

Question

Quizplus · Accepted Answer

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 exists at least one c in (a, b) such that f'(c) = 0. For the function f(x) = 2x^3 - 3x^2, f(-3) = f(3) = -27. Differentiating f(x) gives f'(x) = 6x^2 - 6x. Setting f'(x) = 0 gives x(x - 1) = 0, so x = 0 or x = 1. Only x = 0 is within the interval [-3, 3] and satisfies Rolle's Theorem.

Find the Value C Such That the Conclusion of Rolle's