Find the minimum value of an expression with three variablesHow can I find the minimum of the following expression: ( x y z + z x y + y z x ) ( x y z + y x z + z x y )( x , y , z are non-zero real numbers)The expression can be simplified to ( ( x y ) 2 + ( z x ) 2 + ( y z ) 2 ) ( x 2 + y 2 + z 2 ) ( x y z ) 2 , but I am not sure that it will help anyhow.

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Find the minimum value of an expression with three variablesHow can I find the minimum of the following expression:  (            x      y        z    +            z      x        y    +            y      z        x    )  (      x          y      z        +      y          x      z        +      z          x      y        )(  x  ,  y  ,  z are non-zero real numbers)The expression can be simplified to             (      (      x      y              )        2            +      (      z      x              )        2            +      (      y      z              )        2            )      (              x        2            +              y        2            +              z        2            )              (      x      y      z              )        2            , but I am not sure that it will help anyhow.

espava8b · Accepted Answer

Remember that for any 3 positive numbers we have                    a        +        b        +        c            3        ≥            a      b      c        3  this is inequality between arithmetic mean and geometic mean.Use it twice. First   a  =  (  x  y      )    2  ,   b  =  .  .  .  .and second   a  =      x    2  ,... and you get:  (            x      y        z    +            z      x        y    +            y      z        x    )  (      x          y      z        +      y          x      z        +      z          x      y        )  ≥  9

snaketao0g · Accepted Answer

expanding gives  3  +            x      2              y      2        +            y      2              x      2        +            x      2              z      2        +            z      2              x      2        +            y      2              z      2        +            z      2              y      2      and now use that  A  +      1    A    ≥  2for  A  &amp;gt;  0

Find the minimum value of an expression with three variables How can I find the minimum of the following expression: ((xy)/(z)+(zx)/(y)+(yz)/(x))((x)/(yz)+(y)/(xz)+(z)/(xy))? (x,y,z are non-zero real numbers)

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