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AM-GM Inequality Problem

  1. Apr 23, 2012 #1
    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?
  2. jcsd
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