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\textAnswer = 256 - A - Simpleprint

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 Question: A historian of science is studying the relationship between scientific discoveries and time intervals. Let \( h(x) \) be a polynomial representing a historical model of discoveries. Let \( x, y, z \) be positive real numbers such that \( x + y + z = 6 \). Find the minimum value of \( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \).

📌 To find the minimum value of \( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \), given \( x + y + z = 6 \), use the AM-HM inequality:

📌 \frac{x + y + z}{3} \geq \frac{3}{\frac{1}{x} + \frac{1}{y} + \frac{1}{z}}

📌 Substitute \( x + y + z = 6 \):

📌 \frac{6}{3} \geq \frac{3}{\frac{1}{x} + \frac{1}{y} + \frac{1}{z}}

📌 \geq \frac{3}{\frac{1}{x} + \frac{1}{y} + \frac{1}{z}}

📌 \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \geq \frac{3}{2}

📌 Equality holds when \( x = y = z \). Since \( x + y + z = 6 \), set \( x = y = z = 2 \).

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