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Verify Rolle's theorem for the function f(x) = x2 + 2x - 8, x ∈ [-4, 2]. |
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Answer» f(x) is a polynomial in x. Hence it is continuous over [-4, 2] and differentiable over (-4, 2). f(-4) = (-4)2 + 2 (-4) - 8 = 16 - 8 - 8 = 0 f(2) = 4 + 4 - 8 = 0 ∴ f(-4) = f(2) ∴ All the conditions of the Rolle's theorem are satisfied. ∴ there exists a c ∈[-4, 2] such that f'(c) = 0 f'(x) = 2x + 2 f(c) = 2c + 2 f(c) = 0 ⇒ 2c + 2 = 0 ⇒ 2c = -2 ⇒ c = -1 ∈[-4, 2] ∴ Rolle's theorem is verified. |
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