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Math Help - Minima

  1. #1
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    Minima

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

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

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

    Equality is achieved iff: <br />
x = y = z =1<br />
( 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 P(x, y, z) = (x+y)(y+z)(z+x), when xyz=1 and x, y, z are positive real numbers.
    It looks to be (x,y,z)=(1,1,1), but I'm not sure about that. I certainly couldn't prove it.
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