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Thread: Minima

  1. #1
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    Minima

    Find the minimal value of the product $\displaystyle P(x, y, z) = (x+y)(y+z)(z+x)$, when $\displaystyle xyz=1$ and $\displaystyle x, y, z $ are positive real numbers.
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  2. #2
    Super Member PaulRS's Avatar
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    We have: $\displaystyle
    x + y \ge 2 \cdot \sqrt {xy}
    $ ( just see that $\displaystyle
    \left( {\sqrt x - \sqrt y } \right)^2 \ge 0
    $ (*) and expand)

    Similarly with the others. And multiply: $\displaystyle
    \left( {x + y} \right) \cdot \left( {x + z} \right) \cdot \left( {y + z} \right) \ge 2^3 \cdot \sqrt {xy} \sqrt {xz} \sqrt {yz}
    $

    And note that: $\displaystyle
    \sqrt {xy} \sqrt {xz} \sqrt {yz} = xyz = 1
    $ thus: $\displaystyle
    \left( {x + y} \right) \cdot \left( {x + z} \right) \cdot \left( {y + z} \right) \ge 8
    $

    Equality is achieved iff: $\displaystyle
    x = y = z =1
    $ ( see (*) )
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  3. #3
    Senior Member
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    Quote Originally Posted by Winding Function View Post
    Find the minimal value of the product $\displaystyle P(x, y, z) = (x+y)(y+z)(z+x)$, when $\displaystyle xyz=1$ and $\displaystyle x, y, z $ are positive real numbers.
    It looks to be $\displaystyle (x,y,z)=(1,1,1)$, but I'm not sure about that. I certainly couldn't prove it.
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