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pain in back and under ribs on left side - Decision Point

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 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:

📌 f\left(\frac{\pi}{4}\right) = 1 + \frac{1}{\frac{1}{2}} + \frac{1}{\frac{1}{2}} = 1 + 2 + 2 = 5

📌 Thus, the minimum value of $ f(x) $ is:

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