If the function `f(x)=2tanx+(2a+1)(log)_e|secx|+(a-2)x` is increasing on `R` , then (a) `a in (1//2,\ oo)` (b) `a in (-1//2,\ 1//2)` (c) `a=1//2` (d) `a in R`

Options

A. `(ain (1//2,oo)`
B. `(ain (-1//2,1//2)`
C. `a=1//2`
D. `(ainR)`

Answer

C

Explanation

If `f(x)= tanx+(2a+1)log_e|secx| +(a-2)x` is inreasing on R then
`f'(x) ge 0 " for all " x in R`
`rArr 2sec^2x+(2a+1)tanx+(a-2)ge " for all " x in R`
`rArr 2tan^2x+(2a+1)tanx+age0 " for all " x in R`
`rArr (2a+ 1)^2-8ale 0 rArr (2a-1)^2le 0 rArr a= 1/2`