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Home » Setting each factor to zero gives the roots: - Sourci

Setting each factor to zero gives the roots: - Sourci

📅 April 30, 2026 👤 scraface

Apr 30, 2026

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📌 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:

📌 Question: A science policy analyst is modeling the growth of renewable energy adoption using the function

📌 g(x) = \left(\sqrt{2} \sin x + \frac{1}{\sqrt{2} \sin x}\right)^2

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