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A function f: R -> [-pi/4, pi/2) is defined by f(x) = arctan(x⁴ - x² - 7/4 + arctan(alpha)). If f is surjective (onto), which of the following is correct?

1. cos^(-1)((1 - alpha²)/(1 + alpha²)) = 2
2. alpha + 1/alpha = 2*cosec(2)
3. sin^(-1)(2*alpha/(alpha² + 1)) = pi - 2
4. tan^(-1)(2*alpha/(alpha² - 1)) = 2 - pi

Correct answer: alpha + 1/alpha = 2*cosec(2)

Solution

Surjectivity onto [-pi/4, pi/2) requires the minimum of f to be -pi/4, which forces arctan(alpha) = 1, i.e., alpha = tan(1). Then alpha + 1/alpha = tan(1) + cot(1) = 1/sin(1)cos(1) = 2/sin(2) = 2*cosec(2).

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