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If the minimum and the maximum values of the function f: [π/4, π/2] → R, defined by: f(θ) = | -sin² θ -1 - sin² θ 1 | | -cos² θ -1 - cos² θ 1 | | 12 10 -2 | are m and M respectively, then the ordered pair (m, M) is equal to:

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

Correct answer: (-4, 0)

Solution

Subtracting rows reduces the determinant to 4(cos^2 theta - sin^2 theta) = 4cos(2theta). For theta in [pi/4, pi/2], 2theta in [pi/2, pi], so cos(2theta) in [-1,0] and the value lies in [-4,0]. Hence (m,M) = (-4, 0).

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