(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

{(x+y+z)^3-2(x+y+z)(x^2+y^2+z^2)}/xyz ≤ 9

2. Relevant equations

AM-GM inequality x+y+z ≥ 3(√xyz)(cube root) and xy+yz+zx ≥ 3√(xyz)^2(cube root)

3. The attempt at a solution

This is my attempt but I don't know if I am using the AM-GM inequality correctly

(x+y+z)^3/xyz ≤ 9+ [2(x+y+z)(x^2+y^2+z^2)]/xyz

(x+y+z)^3/xyz ≤ [9xyz+ 2(x+y+z){(x+y+z)(x+y+z)-2(xy+yz+zx)}]/xyz

(3√xyz)3/xyz ≤ [9xyz +2(x+y+z)^3 -4(x+y+z)(xy+yz+zx)]/xyz

27xyz/xyz ≤ [9xyz +2(3√xyz)^3 -4(3√xyz)(3√(xyz)^2)]/xyz

27xyz/xyz ≤ [9xyz +54xyz -36xyz]/xyz

27 ≤ 27

Is this correct?

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# Homework Help: AM-GM Inequality Problem

Can you offer guidance or do you also need help?

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