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Home » Sage Sonic: The Hidden Truth Behind Sonic’s Flashy Secrets You Never Knew! - Baxtercollege

Sage Sonic: The Hidden Truth Behind Sonic’s Flashy Secrets You Never Knew! - Baxtercollege

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 Question: Let $ x, y, z $ be positive real numbers such that $ x + y + z = 1 $. Find the minimum value of $ \frac{1}{x} + \frac{4}{y} + \frac{9}{z} $.

📌 Solution: We apply the **Cauchy-Schwarz inequality** in the Engel form (also known as Titu’s Lemma):

📌 \frac{a_1^2}{b_1} + \frac{a_2^2}{b_2} + \frac{a_3^2}{b_3} \ge \frac{(a_1 + a_2 + a_3)^2}{b_1 + b_2 + b_3}

📌 But our expression is $ \frac{1}{x} + \frac{4}{y} + \frac{9}{z} = \frac{1^2}{x} + \frac{2^2}{y} + \frac{3^2}{z} $

📌 Apply Titu’s Lemma:

📌 \frac{1^2}{x} + \frac

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