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Home » \[ y + 3 = 3x - 6 \] - Dyverse

\[ y + 3 = 3x - 6 \] - Dyverse

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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rac{x^2 + y^2}{xy} + rac{xy}{x^2 + y^2}
subject to $ x, y > 0 $.
Solution: Let $ A = rac{x^2 + y^2}{xy} $ and $ B = rac{xy}{x^2 + y^2...
A + B = rac{x^2 + y^2}{xy} + rac{xy}{x^2 + y^2}.
Let $ t = rac{x^2 + y^2}{xy} $. By the AM-GM inequality,
x^2 + y^2 \geq 2xy \Rightarrow rac{x^2 + y^2}{xy} \geq 2,
so $ t \geq 2 $. The expression becomes $ t + rac{1}{t} $. Define $ f...
f(2) = 2 + rac{1}{2} = rac{5}{2}.
Equality holds when $ x = y $, which satisfies $ x, y > 0 $. Thus, the...
oxed{rac{5}{2}}.

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