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Math Help - Inequality and condition

  1. #1
    Super Member dhiab's Avatar
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    Inequality and condition

    Prove : If
    then :
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  2. #2
    MHF Contributor alexmahone's Avatar
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    (a+b+c)^2 \geq 0

    a^2+b^2+c^2+2ab+2bc+2ca \geq 0

    1+2(ab+bc+ca) \geq 0

    ab+bc+ca \geq -\frac{1}{2}

    -(ab+bc+ca) \leq \frac{1}{2} ----------------- (1)

    ----------------------------------------------------------------------------

    (a-b)^2+(b-c)^2+(c-a)^2=2(a^2+b^2+c^2-ab-bc-ca)

    =2(1-ab-bc-ca)

    \leq 2(1+\frac{1}{2})

    \leq 2(\frac{3}{2})

    \leq 3

    So, (a-b)^2+(b-c)^2+(c-a)^2 \leq 3

    Hence proved.
    Last edited by alexmahone; July 16th 2009 at 05:51 AM.
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