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Find the minimum value of the expression (9*x²*sin²(x) + 4) / (x*sin(x)) for x in (0, pi).

1. 4
2. 6
3. 8
4. 12

Correct answer: 12

Solution

Let t = x*sin(x). For x in (0, pi), sin(x) > 0, so t > 0. The expression is 9t + 4/t. By AM-GM: 9t + 4/t >= 2*sqrt(9t * 4/t) = 2*sqrt(36) = 12. Equality when 9t = 4/t => t² = 4/9 => t = 2/3. The minimum value is 12.

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