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happy days are here again song - IQnection

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Since $ x + y + z = 1 $, we substitute into the inequality:

📌 To check if this bound is achievable, consider the equality condition in Cauchy-Schwarz: equality occurs when

📌 Let this common value be $ k $. Then,

📌 \frac{x^2}{y} = \frac{y^2}{z} \Rightarrow x^2 z = y^3, \quad \frac{y^2}{z} = \frac{z^2}{x} \Rightarrow y^2 x = z^3.

📌 From these, we deduce a symmetry: suppose $ x = y = z $. Then $ x + y + z = 3x = 1 \Rightarrow x = \frac{1}{3} $. Substituting into $ S $:

📌 S = 3 \cdot \frac{(1/3)^2}{1/3} = 3 \cdot \frac{1/9}{1/3} = 3 \cdot \frac{1}{3} = 1.

📌 Hence, the minimum value is achieved when $ x = y = z = \frac{1}{3} $.

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