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📅 May 16, 2026 👤 scraface

May 16, 2026

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📌 Equality holds when $ x = y $, which satisfies $ x, y > 0 $. Thus, the minimum value is

📌 oxed{rac{5}{2}}.

📌 Question: A hydrologist models the flow resistance $ R $ through a porous aquifer as

📌 R = rac{p + q}{p - q} + rac{p - q}{p + q} - 2,

📌 where $ p > q > 0 $ are hydraulic conductivity parameters. Find the minimum value of $ R $.

📌 Solution: Let $ a = rac{p + q}{p - q} $ and $ b = rac{p - q}{p + q} $. Note that $ b = rac{1}{a} $, and since $ p > q > 0 $, $ a > 1 $. Then

📌 R = a + rac{1}{a} - 2.

📌 Define $ f(a) = a + rac{1}{a} - 2 $ for $ a > 1 $. By AM-GM, $ a + rac{1}{a} \geq 2 $, with equality at $ a = 1 $. But $ a > 1 $, so $ f(a) > 0 $. However, as $ a o 1^+ $, $ f(a) o 0^+ $. The minimum value is not attained, but we seek the infimum. Since $ f(a) $ increases for $ a > 1 $, the expression approaches 0 as $ p o q^+ $, but never reaches it. However, we analyze the original expression algebraically:

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