Complete set of solution of `log _(1//3) (2 ^(x +2) - 4 ^(x)) ge -2` is :

Options

A. `(-oo, 2-sqrt(13))`
B. `(-oo, 2+sqrt(13))`
C. `(-oo, 2)`
D. none of these

Answer

C

Explanation

We have,
`"log"_(1//3) (2^(x+2)-4^(x)) ge-2`
`rArr (2^(x+2)-4^(x)) le ((1)/(3))^(-2)`
`rArr 2^(x+2)-(2^(x))^(2) le 9`
`rArr (2^(x))^(2)-4(2^(x))+9 ge 0`, which is true for all ` x in R`.
Now,
`"log"_(1//3) (2^(x+2)-4^(x))` is defined, if
`2^(x+2)-4^(x) gt 0`
`rArr 4(2^(x))-(2^(x))^(2) gt 0`
`rArr 2^(x) (4-2^(x)) gt 0`
`rArr 4-2^(x) gt 0 rArr 2^(x) lt 4 rArr x lt 2 rArr x in (-oo, 2)`
Hence the solution set is `(-oo, 2)`