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).

Sasha Hess 2022-09-12 Answered
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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Answers (2)

darkflamexivcr
Answered 2022-09-13 Author has 14 answers
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 ).

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Konciljev56
Answered 2022-09-14 Author has 2 answers
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'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.

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