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Home » Essayons x = 6 â 2(216) + 8(36) = 432 + 288 = 720 â 384 - AssociationVoting

Essayons x = 6 â 2(216) + 8(36) = 432 + 288 = 720 â 384 - AssociationVoting

📅 May 17, 2026 👤 scraface

May 17, 2026

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📌 This still depends on \( z \). But since the original expression is homogeneous (scaling \( x, y, z \) by a constant does not change the value), we can set \( z = 1 \) without loss of generality.

📌 Let \( z = 1 \), \( y = 1 + b \), \( x = 1 + b + a \), with \( a, b > 0 \). Then:

📌 S = 1 + 2\left( \frac{1 + b}{a} + \frac{1}{b} - \frac{1}{a + b} \right)

📌 Now define:

📌 f(a, b) = \frac{1 + b}{a} + \frac{1}{b} - \frac{1}{a + b}

📌 We seek to minimize \( S = 1 + 2f(a, b) \), so minimize \( f(a, b) \).

📌 Fix \( b \), and minimize over \( a > 0 \). Note that as \( a \to \infty \), \( \frac{1 + b}{a} \to 0 \), so

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