If x + y = a, then the maximum value of x ∗ y is when x = y = a / 2. ( x , y are positive numbers)

Question

If   x  +  y  =  a, then the maximum value of   x  ∗  y is when   x  =  y  =  a      /    2. (  x  ,  y are positive numbers)

persstemc1 · Accepted Answer

x  y  =            (      x      +      y              )        2            −      (      x      −      y              )        2              4    =                    a        2            −      (      x      −      y              )        2              4    ≤            a      2        4  with equaloity iff   x  −  y  =  0

Sylvia Byrd · Accepted Answer

With  y  =  a  −  xwe get  x  (  a  −  x  )  =  a  x  −      x    2    =  −      (          x      2        −    a    x    +                  a        2            4        −                  a        2            4        )    =  −            (      x      −              a        2            )        2    +            a      2        4    ≤            a      2        4

If x + y = a , then the maximum value of x ∗ y is when

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