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Home » If \( y = 0 \), \( 2x = 120 \rightarrow x = 60 \) - Esdistancia

If \( y = 0 \), \( 2x = 120 \rightarrow x = 60 \) - Esdistancia

📅 March 17, 2026 👤 scraface

Mar 17, 2026

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📌 f(n) = \frac{n^2 + 2n + 3}{n + 1}.

📌 Perform polynomial division:

📌 f(n) = n + 1 + \frac{2}{n + 1}.

📌 Let $ x = n + 1 $, so $ x > 1 $, and $ f(n) = x + \frac{2}{x} $. Define $ g(x) = x + \frac{2}{x} $. Take derivative:

📌 g'(x) = 1 - \frac{2}{x^2}.

📌 Set $ g'(x) = 0 $:

📌 - \frac{2}{x^2} = 0 \Rightarrow x^2 = 2 \Rightarrow x = \sqrt{2}.

📌 Since $ g''(x) = \frac{4}{x^3} > 0 $ for $ x > 0 $, this is a minimum. Thus, the minimum value of $ f(n) $ is $ \sqrt{2} + \frac{2}{\sqrt{2}} = \sqrt{2} + \sqrt{2} = 2\sqrt{2} $.

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Question: A hydrologist models the flow resistance $ R $ through a por...
R = rac{p + q}{p - q} + rac{p - q}{p + q} - 2,
where $ p > q > 0 $ are hydraulic conductivity parameters. Find the mi...
Solution: Let $ a = rac{p + q}{p - q} $ and $ b = rac{p - q}{p + q}...
R = a + rac{1}{a} - 2.
Define $ f(a) = a + rac{1}{a} - 2 $ for $ a > 1 $. By AM-GM, $ a + r...
R = rac{(p+q)^2 + (p-q)^2}{(p-q)(p+q)} - 2 = rac{p^2 + 2pq + q^2 + p...
R = 2\left( rac{p^2 + q^2}{p^2 - q^2} - 1
ight) = 2\left( rac{2q^2}{p^2 - q^2}
ight) = rac{4q^2}{p^2 - q^2}.

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