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Home » Relax and Warm Up with the Ultimate Hug GIF – Feel the Love Instantly! - Crosslake

Relax and Warm Up with the Ultimate Hug GIF – Feel the Love Instantly! - Crosslake

📅 March 1, 2026 👤 scraface

Mar 01, 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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