If `"log"_(sqrt(3))("sin" x + 2sqrt(2) "cos"x) ge 2, -2pi le x le 2 pi,` then the number of values of x, is

Options

A. 0
B. 3
C. infinite
D. none of these

Answer

D

Explanation

We have,
`"log"_(sqrt(3)) ("sin" x + 2sqrt(2) "cos" x) ge 2`
`rArr ("sin" x + 2 sqrt(2) "cos"x) ge (sqrt(3))^(2)`
`rArr "sin"x +2 sqrt(2) "cos" x ge 3`
`rArr "sin" x + 2sqrt(2) "cos" x = 3 " " [-3 le "sin"x + 2sqrt(2) "cos" x le 3]`
`rArr "sin" (x + theta)= 1, "where cos" theta = 1//3`
`rArr x + theta = n pi + (-1)^(n) (pi)/(2) rArr x = n pi + (-1)^(n) (pi)/(2)-"cos"^(-1) ((1)/(3))`
Clearly, `x in [-2pi, 2pi], "if"n = 0, 1, -1, -2.`