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Math Help - Equality

  1. #1
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    Equality

    For non-zero a,b,c

    a^2+b^2+c^2+\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c  ^2}\geq 6

    show also that equality holds if and only if a,b,c\in\{-1,1\}.

    How can I show this?
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  2. #2
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    Quote Originally Posted by dwsmith View Post
    For non-zero a,b,c
    a^2+b^2+c^2+\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c  ^2}\geq 6
    show also that equality holds if and only if a,b,c\in\{-1,1\}.
    That "only if" part is not true.
    \left( {\forall x,y} \right)\left[ {x^2  + y^2  \geqslant 2xy} \right].
    If t\ne 0\text{ let }x=t~\&~y=\frac{1}{t},~t=a,b,c.
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  3. #3
    MHF Contributor alexmahone's Avatar
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    Quote Originally Posted by dwsmith View Post
    For non-zero a,b,c

    a^2+b^2+c^2+\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c  ^2}\geq 6

    show also that equality holds if and only if a,b,c\in\{-1,1\}.

    How can I show this?
    a^2+\frac{1}{a^2}\geq2

    b^2+\frac{1}{b^2}\geq2

    c^2+\frac{1}{c^2}\geq2

    (Using AM-GM inequality three times)

    Adding,

    a^2+b^2+c^2+\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c  ^2}\geq 6

    Each of the AM-GM inequalities becomes an equality if and only if a^2=\frac{1}{a^2}, b^2=\frac{1}{b^2}, c^2=\frac{1}{c^2}

    ie a=\pm1, b=\pm1, c=\pm1

    ie a,b,c\in\{-1,1\}
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