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Home » Let the width be \( w \) meters. - Aurero

Let the width be \( w \) meters. - Aurero

📅 March 13, 2026 👤 scraface

Mar 13, 2026

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📌 D(x) = x^2 + \left( \frac{3}{4}x + 2 \right)^2

📌 D(x) = x^2 + \frac{9}{16}x^2 + 3x + 4 = \frac{25}{16}x^2 + 3x + 4

📌 D'(x) = \frac{50}{16}x + 3 = \frac{25}{8}x + 3 = 0 \Rightarrow x = -\frac{24}{25}

📌 Substitute into the line equation to find $ y $:

📌 y = \frac{3}{4} \cdot \left( -\frac{24}{25} \right) + 2 = -\frac{18}{25} + 2 = \frac{32}{25}

📌 Thus, the closest point is $ \left( -\frac{24}{25}, \frac{32}{25} \right) $.

📌 \boxed{\left( -\frac{24}{25}, \frac{32}{25} \right)}

📌 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) $.

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