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📅 May 17, 2026 👤 scraface

May 17, 2026

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📌 Equality occurs when $ \frac{x}{y} = \frac{y}{z} = \frac{z}{x} $, which implies $ x = y = z $. Since $ x + y + z = 1 $, we have $ x = y = z = \frac{1}{3} $.

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

📌 Therefore, the minimum value is $ \boxed{1} $.

📌 Question: Compute the sum of the roots of the equation

📌 u\sqrt{u} - 6u + 7\sqrt{u} - 12 = 0,

📌 given that all roots are real and non-negative.

📌 Solution: Let $ \sqrt{u} = t $, so $ u = t^2 $. Substituting into the equation:

📌 t^2 \cdot t - 6t^2 + 7t - 12 = 0 \Rightarrow t^3 - 6t^2 + 7t - 12 = 0

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