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Home » Only positive value: \( x = 2 + 2\sqrt5 \) - Blask

Only positive value: \( x = 2 + 2\sqrt5 \) - Blask

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Using the Pythagorean identity $ \sin^2 x + \cos^2 x = 1 $, this becomes:

📌 f(x) = 1 + \frac{1}{\sin^2 x} + \frac{1}{\cos^2 x}

📌 Let $ a = \sin^2 x $, $ b = \cos^2 x $, so $ a + b = 1 $, and $ a, b > 0 $ in $ (0, \frac{\pi}{2}) $. Then:

📌 f(x) = 1 + \frac{1}{a} + \frac{1}{b}

📌 We now minimize $ \frac{1}{a} + \frac{1}{b} $ under the constraint $ a + b = 1 $. Using the method of Lagrange multipliers or AM-HM inequality:

📌 \frac{1}{a} + \frac{1}{b} \geq \frac{4}{a + b} = 4

📌 Equality occurs when $ a = b = \frac{1}{2} $, i.e., when $ \sin^2 x = \cos^2 x = \frac{1}{2} $, or $ x = \frac{\pi}{4} $.

📌 Substituting back:

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