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For 0 < theta < pi, find the minimum value of the expression f(theta) = 3*sin(theta) + csc³(theta).

1. 4
2. 3
3. 5
4. 6

Correct answer: 4

Solution

Let t = sin(theta). Since 0 < theta < pi, t in (0, 1]. f = 3t + t⁻³. Differentiate: df/dt = 3 - 3t⁻⁴. Set to zero: 3 = 3t⁻⁴ => t⁴ = 1 => t = 1. Second derivative: 12t⁻⁵ > 0 confirms minimum. At t = 1 (theta = pi/2): f = 3*1 + 1/1³ = 4.

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