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Home » The perimeter is \( 2(w + 2w + 3) = 54 \). - Portal da AcÃºstica

The perimeter is \( 2(w + 2w + 3) = 54 \). - Portal da AcÃºstica

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Question: A cloud computing consultant models data flow efficiency as $ f(x) = (\sin x + \cos x)^2 + \frac{1}{\sin x \cos x} $, for $ 0 < x < \frac{\pi}{2} $. Find the minimum value of $ f(x) $.

📌 Solution: First, expand $ (\sin x + \cos x)^2 = \sin^2 x + 2\sin x \cos x + \cos^2 x = 1 + 2\sin x \cos x $.

📌 Let $ s = \sin x \cos x $. Then:

📌 f(x) = 1 + 2s + \frac{1}{s}

📌 Note $ s = \frac{1}{2} \sin 2x $, and since $ 0 < x < \frac{\pi}{2} $, $ 0 < 2x < \pi $, so $ 0 < \sin 2x \leq 1 $, hence $ 0 < s \leq \frac{1}{2} $.

📌 Define $ g(s) = 1 + 2s + \frac{1}{s} $, for $ 0 < s \leq \frac{1}{2} $.

📌 g'(s) = 2 - \frac{1}{s^2}

📌 Set $ g'(s) = 0 \Rightarrow 2 = \frac{1}{s^2} \Rightarrow s^2 = \frac{1}{2} \Rightarrow s = \frac{1}{\sqrt{2}} \approx 0.707 > \frac{1}{2} $, not in domain.

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