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

Timiavawsw9

Timiavawsw9

Answered question

2022-05-21

Inequality with the constraint x y = x + y
I 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.

Answer & Explanation

Keyon Fitzgerald

Keyon Fitzgerald

Beginner2022-05-22Added 10 answers

Since ( x + y ) 2 4 x y = x + y , we obtain x + y 4
Thus, 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

Denisse Valdez

Beginner2022-05-23Added 5 answers

By renaming x as 1 u and y as 1 v we have to prove that u , v > 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.

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