`e^(|sinx|)+e^(-|sinx|)+4a=0` will have exactly four different solutions in `[0,2pi]` if. `a in R` (b) `a in [-3/4,-1/4]` `a in [(-1-e^2)/(4e),oo]` (d) none of these

Options

A. `a in R`
B. `a in [- e/4, -1/4]`
C. `a in [(-1-e^(2))/(4e), oo]`
D. none of these

Answer

D

Explanation

`e^(|sin x|)+e^(-|sin x|)+4a=0`. Let `t=e^(|sin x|)`
`rArr t in [1, e]`
`rArr t+1/t +4a=0`
`rArr t+1/t=-4a`
Let `f(t)=t+1/t`
`:. F'(t)=1-1/t^(2) ge 0` for `t in [1, e]`
So, `f(t)` is increasing function for `t in [1, e]`
`:. f(t) in [f(1), f(e)]`
or `f(t) in [2, (e^(2)+1)/e]`
`:. -4a in [2, (e^(2)+1)/e]`
`:. a in [(-(e^(2)+1))/(4e), (-1)/2]`