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doubletree hilton ocean point north miami beach - ECD Germany

📅 May 5, 2026 👤 scraface

May 05, 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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