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Let p, q, r be real numbers. Define f(x) as the determinant: | x+p² pq pr | | pq x+q² qr | | pr qr x+r²| Find the intervals where f(x) is increasing and where it is decreasing.

1. f increases for x < -2/3*(p² + q² + r²) and x > 0
2. f decreases for x in (-2/3*(p² + q² + r²), 0)
3. f decreases for x < -2/3*(p² + q² + r²) and x > 0
4. f increases for x in (-2/3*(p² + q² + r²), 0)

Correct answer: f increases for x < -2/3*(p² + q² + r²) and x > 0

Solution

The matrix equals x*I + v*v^T (rank-1 update), so det = x³ + x²*S = x²*(x+S). Then f'(x) = x*(3x+2S). Roots at x=0 and x=-2S/3. f' > 0 (increasing) when x < -2S/3 or x > 0; f' < 0 (decreasing) when -2S/3 < x < 0. Both options A and B are correct.

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