If `x ,y ,z` are positive numbers is `AdotPdot,t h e n` `y^2geqx z` `x y+y zgeq2x z` `(x+y)/(2y-x)+(y+z)/(2y-z)` d. none of these

Options

A. `y^2ge xz`
B. `xy+yzge 2xz`
C. `(x+y)/(2y-x)+(y+z)/(2y-z)ge4`
D. none of these

Answer

A, C

Explanation

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]`