If G and L are the greatest and least values of the expression `(x^(2)-x+1)/(x^(2)+x+1), x epsilon R` respectively. then find the least value of G^(5)+L^(5)` .

Options

A. 0
B. 2
C. 16
D. 32

Explanation

Let `y=(x^(2)-x+1)/(x^(2)+x+1)`
`impliesx^(2)y+xy+y=x^(2)-x+1`
`implies(y-1)x^(2)+(y+1)x+y-1=0 [ :' x epsilonR]`
`:.(y+1)^(2)-4 . (y-1)(y-1)ge0 [ :' b^(2)-4acge0]`
`implies(y+1)^(2)-(2y-2)^(2)ge0`b
`implies(3y-1)(y-3)le0`
`:.1/3leyle3impliesG=3` and `L=1/3:.GL=1`
`(G^(5)+L^(5))/2ge(GL)^(1//5)=(1)^(1//5)=1`
`implies(G^(5)+L^(5))/2 ge 1` or `G^(5)+L^(5)ge2`
`:.` Minimum value of `G^(5)+L^(5)` is 2.