If G and L are the greatest and least values of the expression`(2x^(2)-3x+2)/(2x^(2)+3x+2), x epsilonR` respectively.
The least value of `G^(100)+L^(100)` is

Options

A. `2^(100)`
B. `3^(100)`
C. `7^(100)`
D. none of these

Answer

D

Explanation

Let `y=(2x^(2)-3x+2)/(2x^(2)+3x+2)`
`implies2x^(2)y+3xy+2y=2x^(2)-3x+2`
`implies2(y-1)x^(2)+3(y+1)x+2(y-1)=0`
As `x epsilonR`
`:.Dge0`
`=9(y+1)^(2)-4.2(y-1).2(y-1)ge0`
`implies9(y+1)^(2)-16(y-1)^(2)ge0`
`implies(3y+3)^(2)-(4y-4)^(2)ge0`
`implies(7y-1)(7-y)ge0`
`implies(7y-1)(y-7)le0`
`:.1/7leyle7`
`:.G=7` and `L=1/7`
`:.GL=1`
NOw `(G^(100)+L^(100))/2ge(GL)^(100)implies(G^(100)+L^(100))/2ge1`
`impliesG^(100)+L^(100)ge2`
Least value of `G^(100)` +L^(100)` is 2.