If the function f(x)=3 cos |x| -6 ax +b increases for all `x in R` then the range of value of a given by

Options

A. `(-1/2, oo)`
B. `(-oo,-1//2)`
C. `(-oo,-2)`
D. `(-2 ,oo)`

Answer

B

Explanation

We have
f(x)=3 cos |x|-6ax +b
`rArr f(x) =-3 sin x -6ax +b`
`rArr f(x) =3 sin -6 ax + b`
`[ because cos |x| =cos x for all x ]`
`rArr f(x)=-3 sinx -6a`
For f(x) to be increasing on R we must have
`f(x) gt 0 "for all" n in R`
`rArr -3 sin x-6 a gt 0 "for all"x in R`
`rArr sin x+2a lt 0 "for all x in R`
`rArr sin x lt -2a " for all " x in R`
`rArr 1 lt - 2a " "[ because "`Max .value of sin x is 1]
`rArr a lt -1/2`
`rArr a in (-oo,-1 //2)`