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**emakarov** The function sin^2(x) assumes all values between 0 and 1 and only them. Therefore, the minimum of 4sin^2(x) + 9/sin^2(x) on all real line equals the minimum of 4x + 9/x when 0 < x <= 1. Moreover, the minimum of 4x + 9/x on this interval is assumed when x = 1, so the minimum of 4sin^2(x) + 9/sin^2(x) is assumed when sin^2(x) = 1. If you need more explanation, please say if this is not intuitively clear or if you are looking for formal reasons, such as how this can be deduced from the axioms of real numbers. (Deduction from axioms, even though it is the most air-tight proof, is often not the clearest.)

What is the difference?