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Home » Avoid Cheap Stones—Why This Moissanite Chain Is a Smart Luxury Investment! - Groen Casting

Avoid Cheap Stones—Why This Moissanite Chain Is a Smart Luxury Investment! - Groen Casting

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 g(x) = \left(\sqrt{2} \sin x + \frac{1}{\sqrt{2} \sin x}\right)^2 = 2 \sin^2 x + 2 + \frac{1}{2 \sin^2 x}

📌 Let $ y = \sin^2 x $, where $ y > 0 $. Then:

📌 g(x) = 2y + 2 + \frac{1}{2y}

📌 We minimize $ 2y + \frac{1}{2y} + 2 $. Let:

📌 h(y) = 2y + \frac{1}{2y}

📌 Using the AM-GM inequality:

📌 2y + \frac{1}{2y} \geq 2\sqrt{2y \cdot \frac{1}{2y}} = 2

📌 Equality holds when $ 2y = \frac{1}{2y} \Rightarrow 4y^2 = 1 \Rightarrow y = \frac{1}{2} $.

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