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If x,y,z are positive numbers in A.P., t...

If x,y,z are positive numbers in A.P., then

A

`y^2ge xz`

B

`xy+yzge 2xz`

C

`(x+y)/(2y-x)+(y+z)/(2y-z)ge4`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
A, C
A.M. of x and z is y G.M of x and z is `sqrt(xz)`
Now,`A.M ge G.M implies y^(2) ge xz`
Also, `A.M ge H.M implies y ge (2xz)/(x + z)`
`(x + y)/(2y - x) = (x + y)/(x + z - x) = (x + y)/(z)` and `(y + z)/(2y - z)= (y + z)/(x)`
`:. ((x + y)/(2y - x) + (y + z)/(2y - z))/(2) ge sqrt((x + y)/(z)(y + z)/(x))`
`= sqrt(1 + (y(x + y + z))/(xz)) [:' x + z = 2y]`
`= sqrt(1 + (2y^(2))/(xz))`
`:. (x + y)/(2y - z) + (y + z)/(2y - z) ge 2 sqrt(1 + 3 (y^(2))/(xz)) ge 4 " "[:' y^(2) ge xz]`
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