How to maximize f ( x , y ) = x + y − 2 x y ?How to maximize f ( x , y ) = x + y − 2 x y where x , y ∈ { 1 , 2 , … , n }?It seems that maximum will occur when ( x , y ) = ( 1 , n ) or ( n , 1 ) .

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How to maximize   f  (  x  ,  y  )  =            x      +      y      −      2              x      y      ?How to maximize   f  (  x  ,  y  )  =            x      +      y      −      2              x      y       where   x  ,  y  ∈  {  1  ,  2  ,  …  ,  n  }?It seems that maximum will occur when   (  x  ,  y  )  =  (  1  ,  n  ) or   (  n  ,  1  )  .

darkflamexivcr · Accepted Answer

Note that   f  (  1  ,  1  )  =  0 and   f  (  1  ,  2  )  =  f  (  2  ,  1  )  =      1    2  Otherwise,                    f        (        x        ,        y        )                            =                              x            +            y            −            2                                x            y                                                            =                  1          y                +                  1          x                −                  2                      x            y                                                            =                  1          y                          (          1          −                      1            x                    )                +                  1          x                          (          1          −                      1            y                    )                                                  ≤                  1          2                          (          1          −                      1            x                    )                +                  1          2                          (          1          −                      1            y                    )                                                  ≤        1        −                  1          n                    Note that  f  (  1  ,  n  )  =  1  −      1    n  Thus the maximum is   1  −      1    n   which is achieved at   (  1  ,  n  ).

Konciljev56 · Accepted Answer

Let M be a maximal value.Thus,  M  x  y  −  x  −  y  +  2  ≥  0  ,which is a linear inequality of x and of y, which says that it&#039;s enough to check this inequality for the extreme values of x and y:  (  x  .  y  )  ∈  {  (  1  ,  1  )  ,  (  n  ,  n  )  ,  (  1  ,  n  )  ,  (  n  ,  1  )  }  ,which gives  M  ∈      {    0    ,                  2        (        n        −        1        )                    n        2              ,                  n        −        1            n        }    ,which gives that             n      −      1        n   is a maximal value.

How to maximize f(x,y)=(x+y-2)/(xy) where x,y in {1,2,…,n}? It seems that maximum will occur when (x,y)=(1,n) or (n,1).

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