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Home » Solution: The arithmetic mean is the sum divided by 3: - Midis

Solution: The arithmetic mean is the sum divided by 3: - Midis

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 But the problem likely expects a finite maximum — perhaps constraint misunderstanding.

📌 Wait — re-express:

📌 Let \(t = \sin x \cos x = \frac{1}{2} \sin 2x\), so for \(x \in (0, \frac{\pi}{2})\), \(t \in (0, \frac{1}{2}]\).

📌 f(x) = \frac{1}{t^2} + \frac{2}{t} = \frac{1 + 2t}{t^2}

📌 Let \(h(t) = \frac{1 + 2t}{t^2}\), \(t \in (0, \frac{1}{2}]\)

📌 As \(t \to 0^+\), \(h(t) \to \infty\), and \(h'(t) = \frac{2t^2 - (1 + 2t)(2t)}{t^4} = \frac{2t - 2(1 + 2t)}{t^3} = \frac{2t - 2 - 4t}{t^3} = \frac{-2t - 2}{t^3} < 0\), so decreasing.

📌 Maximum value is approached as \(t \to 0^+\), i.e., as \(x \to 0^+\) or \(\frac{\pi}{2}^-\), and \(

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