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Math Help - proof of inequality

  1. #1
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    proof of inequality

    please\,\, have \,\,a \,\,try !

    (1) a+b+c=1

    (2)a,b,c \in R^+

    prove :\,\,27(a^3+b^3+c^3)\geq 6(a^2+b^2+c^2)+1

    with \,\,greatest \,\, appreciation \,\,! !
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: proof of inequality

    Quote Originally Posted by albert940410 View Post
    (1) a+b+c=1 (2)a,b,c \in R^+ prove :\,\,27(a^3+b^3+c^3)\geq 6(a^2+b^2+c^2)+1
    One way: consider f(x,y,z)=27(x^3+y^3+z^3)-6(x^2+y^2+z^2)-1 defined on the compact set

    K=\{(x,y,z)\in\mathbb{R}^3:x+y+z+1=0,\;x\geq 0,\;y\geq 0,\;z\geq 0\}

    Prove that the absolute minimum of f is 0 .
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  3. #3
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    Re: proof of inequality

    You could use Lagrange multipliers or a similar technique, but I'm pretty sure this can be solved without calculus (e.g. Muirhead's inequality or AM-GM).
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