SOLUTION: if x and y are positive real numbers, use the fact that t+(1/t)>=2 for positive numbers to show that (x+y)(xy+1)>=4xy.

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: if x and y are positive real numbers, use the fact that t+(1/t)>=2 for positive numbers to show that (x+y)(xy+1)>=4xy.      Log On


   

Question 26172: if x and y are positive real numbers, use the fact that t+(1/t)>=2 for positive numbers to show that (x+y)(xy+1)>=4xy.
Answer by venugopalramana(3286) About Me  (Show Source):
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if x and y are positive real numbers, use the fact that t+(1/t)>=2 for positive numbers to show that
TST. (x+y)(xy+1)>=4xy.
SINCE X AND Y ARE POSITIVE DIVIDE BOTH SIDES BY XY
TST. (x+y)(xy+1)/XY>=4
TST. {(x+y)/X}{(xy+1)/Y}>=4
LHS=(1+Y/X)(X+1/Y)
=X+1/Y+Y+1/X=(X+1/X)+(Y+1/Y)>=2+2=4
AS PER THE GIVEN RELATION
t+(1/t)>=2