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Home » La longitud es \( 3 \times 8 = 24 \, \textm \). - Silent Sales Machine

La longitud es \( 3 \times 8 = 24 \, \textm \). - Silent Sales Machine

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 #### 6.27Question: Let $ x, y, z $ be positive real numbers such that $ x + y + z = 1 $. Find the minimum value of the expression

📌 Solution: To minimize the expression

📌 S = \frac{x^2}{y} + \frac{y^2}{z} + \frac{z^2}{x},

📌 given that $ x + y + z = 1 $ and $ x, y, z > 0 $, we apply the **Cauchy-Schwarz Inequality** in the following form:

📌 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,

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