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Let f(x) = | 1+sin^2 x cos^2 x sin 2x | | sin^2 x 1+cos^2 x sin 2x | | sin^2 x cos^2 x 1+sin 2x |, x ∈ [π/6, π/3]. If α and β respectively are the maximum and minimum values of f, then

1. (1) β^2 + 2√α = 19/4
2. (2) α^2 + β^2 = 9/2
3. (3) α^2 − β^2 = 4√3
4. (4) β^2 − 2√α = 19/4

Correct answer: (4) β^2 − 2√α = 19/4

Solution

The correct option relates the minimum value β and the maximum value α of the function f(x) in a way that balances their contributions, specifically showing that the difference between the square of the minimum and twice the square root of the maximum equals a constant. This relationship is consistent with the behavior of the function over the specified interval.

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