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Home » 5a + b = 7 \quad \text(Equation 5) - Baxtercollege

5a + b = 7 \quad \text(Equation 5) - Baxtercollege

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Question: Let $ a $, $ b $, and $ c $ be positive real numbers such that $ a + b + c = 1 $. Find the minimum value of $ rac{1}{a} + rac{1}{b} + rac{1}{c} $.

📌 Solution: We are to minimize $ rac{1}{a} + rac{1}{b} + rac{1}{c} $ under the constraint $ a + b + c = 1 $ with $ a, b, c > 0 $.

📌 By the AM-HM inequality:

📌 rac{a + b + c}{3} \geq rac{3}{rac{1}{a} + rac{1}{b} + rac{1}{c}} \Rightarrow rac{1}{3} \geq rac{3}{rac{1}{a} + rac{1}{b} + rac{1}{c}}

📌 Taking reciprocals (since all terms are positive):

📌 rac{1}{a} + rac{1}{b} + rac{1}{c} \geq 9

📌 Equality occurs when $ a = b = c = rac{1}{3} $. Thus, the minimum value is:

📌 Question: Let $ h(x) $ be a polynomial such that $ h(x^3 + 1) = x^6 + 3x^3 + 3 $. Find $ h(x^3 - 1) $.

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