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how many mission impossible movies are there - Malaeb

📅 May 5, 2026 👤 scraface

May 05, 2026

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📌 Solution: We are to minimize the expression

📌 subject to $ x + y + z = 1 $, with $ x, y, z > 0 $.

📌 Apply the **Cauchy-Schwarz Inequality** in the following form:

📌 \left( \frac{x^2}{y} + \frac{y^2}{z} + \frac{z^2}{x} \right)(y + z + x) \geq (x + y + z)^2

📌 Since $ x + y + z = 1 $, we have:

📌 \left( \frac{x^2}{y} + \frac{y^2}{z} + \frac{z^2}{x} \right)(1) \geq 1^2 = 1

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

📌 Equality holds when $ \frac{x}{y} = \frac{y}{z} = \frac{z}{x} $, which implies $ x = y = z $. Since $ x + y + z = 1 $, we get $ x = y = z = \frac{1}{3} $.

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