Inequality with the constraint x y = x + yI have to prove that if x and y are two positive numbers such that x y = x + y, the following inequality x y 2 + 4 + y x 2 + 4 ≥ 1 2 holds. Any help is appreciated, thank you in advance.

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Inequality with the constraint   x  y  =  x  +  yI have to prove that if   x and   y are two positive numbers such that   x  y  =  x  +  y, the following inequality      x                  y        2            +      4        +      y                  x        2            +      4        ≥      1    2  holds. Any help is appreciated, thank you in advance.

Keyon Fitzgerald · Accepted Answer

Since             (      x      +      y              )        2              4    ≥  x  y  =  x  +  y , we obtain   x  +  y  ≥  4Thus, by C-S       x                  y        2            +      4        +      y                  x        2            +      4        ≥            (      x      +      y              )        2                    x              y        2            +              x        2            y      +      4      (      x      +      y      )        =            x      +      y              x      y      +      4        =      1          1      +              4                  x          +          y                      ≥      1    2

Denisse Valdez · Accepted Answer

By renaming   x as       1    u   and   y as       1    v   we have to prove that   u  ,  v  &amp;gt;  0 and   u  +  v  =  1 grant            u      2              v      +      4              u        2            v        +            v      2              u      +      4      u              v        2              ≥      1    2  that is the same as showing that for any   x  ∈  (  0  ,  1  ) the inequality            x      2              (      1      −      x      )      (      1      +      4              x        2            )        +            (      1      −      x              )        2                    x      (      1      +      4      (      1      −      x              )        2            )        ≥      1    2  holds. That is almost trivial since   f  (  x  )  =            x      2              (      1      −      x      )      (      1      +      4              x        2            )       is a convex function on   (  0  ,  1  ), hence  f  (  x  )  +  f  (  1  −  x  )  ≥  2  ⋅  f      (          1      2        )    =      1    2  as wanted.

Inequality with the constraint x y = x + y I have to prove that if x and

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