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Home » Gilmer, TX: Why Everyone’s Talking About This Tiny Town Changing the Game! - High Altitude Science

Gilmer, TX: Why Everyone’s Talking About This Tiny Town Changing the Game! - High Altitude Science

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 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.

📌 So minimum occurs at endpoint $ s = \frac{1}{2} $:

📌 g\left( \frac{1}{2} \right) = 1 + 2 \cdot \frac{1}{2} + \frac{1}{\frac{1}{2}} = 1 + 1 + 2 = 4

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