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f(ab) = f(a)f(b) - 500apps

📅 March 13, 2026 👤 scraface

Mar 13, 2026

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g(x) = \left(\sqrt{2} \sin x + \frac{1}{\sqrt{2} \sin x}\right)^2
Find the minimum value of $ g(x) $ for $ x $ such that $ \sin x > 0 $.
We expand the expression:
g(x) = \left(\sqrt{2} \sin x + \frac{1}{\sqrt{2} \sin x}\right)^2 = 2...
Let $ y = \sin^2 x $, where $ y > 0 $. Then:
g(x) = 2y + 2 + \frac{1}{2y}
We minimize $ 2y + \frac{1}{2y} + 2 $. Let:
h(y) = 2y + \frac{1}{2y}
Using the AM-GM inequality:
2y + \frac{1}{2y} \geq 2\sqrt{2y \cdot \frac{1}{2y}} = 2

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