Just an inequality

Algebra Level 5

( 1 − x ) ( x − y + 1 ) ( x + y ) \large (1-x)\sqrt{(x-y+1)(x+y)} Let x,y be positive reals satisfying − x ≤ y ≤ x + 1 ≤ 2 -x\leq y\leq x+1\leq2 Find the product of x and y satisfied the maximum value of the expression above


The answer is 0.125.

P C by P C , 20, Vietnam

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1 solution

P C
Jan 16, 2016

C can be written as C = ( 1 − x ) 2 ( x − y + 1 ) ( x + y ) C=\sqrt{(1-x)^2(x-y+1)(x+y)} Using AM-GM we get C ≤ ( 1 − x + 1 − x + x − y + 1 + x + y 4 ) 2 C\leq\bigg(\frac{1-x+1-x+x-y+1+x+y}{4}\bigg)^2 ⇔ C ≤ 9 16 \Leftrightarrow C\leq\frac{9}{16} The equality holds when 1 − x = x − y + 1 = x + y 1-x=x-y+1=x+y ⇔ x = 1 4 ; y = 1 2 \Leftrightarrow x=\frac{1}{4};y=\frac{1}{2} ⇒ x . y = 1 8 = 0.125 \Rightarrow x.y=\frac{1}{8}=0.125

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