`f(x)=log_(5)(cos^(-1)sqrt(x^(2)+5x+6)+sec^(-1)[{x}+1]([.]` denotes greatest integer function and `{.}` denotes fractional part function. Find the domain of `f(x)`:

Options

A. `[(-5-sqrt(5))/(2),-3]cup[-2,(-5+sqrt(5))/(2)]`
B. `[(-sqrt(5)+5)/(2),-3]cup[-2,(5+sqrt(5))/(2))`
C. `((-5+sqrt(5))/(4),-3]cup[-2,(-5+sqrt(5))/(4))`
D. `((-5-sqrt(5))/(2),-3]cup[-2,(-5+sqrt(5))/(2))`

Answer

D

Explanation

Domain `sec^(-1)[{x}+1]` is R
`x^(2)+5x+6=0` at `x=-2,-3`
`x^(2)+5x+6=1atx=(-5sqrt(5))/(2).(-5sqrt(5))/(2)`
Domain of `f(x)` is `((-5-sqrt(5))/(2),-3]cup[-2,(-5+sqrt(5))/(2))`