The inequality `sqrt(x^((log)_2sqrt(x)))geq2` is satisfied (a) by only one value of `x` (b) `x in (0,(1/4))` (c)`x in [4,oo)` (d) `x in (1,2)`

Options

A. only one value of x
B. ` x in (0, 1/4]`
C. ` x in [4, infty)`
D. ` x in (1, 2)`

Answer

B, C

Explanation

` sqrt(x^(log_(2)sqrtx))ge 2, x gt 0`
` rArr x^(log_(2)sqrtx) ge 4`
Taking log on both sides with base 2, we get
` log_(2)x^(log_(2)sqrtx)ge 2`
` or (log_(2)sqrtx)(log_(2) x) ge 2`
` or (1//2)(log_(2)x)(log_(2)x) ge 2`
` or (log_(2)x)^(2) ge 4`
` or (log_(2)x-2)(log_(2)x+2) ge 0`
` rArr log_(2)x le - 2 or log_(2)x ge 2`
` rArr x le 1//4 or x ge 4`
` rArr x in (0, 1//4] cup [4, infty)`