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Find the pair (a, b) with b > a > -1 for which the definite integral I = integral from a to b of (x³ - x² - 2x) dx is minimized.

1. (-1, 2)
2. (-1, 0)
3. (0, 2)
4. (-1, 1)

Correct answer: (0, 2)

Solution

The integrand x³ - x² - 2x = x(x-2)(x+1) has roots at -1, 0, 2. Among intervals with b > a > -1, only (0, 2) lies in the region where f(x) < 0, making the integral most negative (minimum). Intervals starting at -1 are excluded since a must be strictly greater than -1.

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