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Question 8.4.3: Test the mean value theorem on the function f (x) = x³ − x, ......

Test the mean value theorem on the function f (x) = x³ − x, defined over [0, 2].

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We find that [f (2) − f (0)]/(2 − 0) = 3 and f^{\prime}(x) = 3x^{2} − 1. The equation f^{\prime}(x) = 3 has two solutions, x = ±2\sqrt{3}/3. The positive root x^{∗} = 2\sqrt{3}/3 ∈ (0, 2), and

f^{\prime}(x^{*})={\frac{f(2)-f(0)}{2-0}}

This confirms the mean value theorem in this case.

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