The range of the function `tan^(-1)((x^2+1)/(x^2+sqrt(3))),x in R` is a.`[pi/6,pi/2)` b. `[pi/6,pi/3)` c. `[pi/6,pi/4]` d. none of these

Options

A. `[(pi)/(6),(pi)/(2))`
B. `[(pi)/(6),(pi)/(3))`
C. `[(pi)/(6),(pi)/(4))`
D. none of these

Answer

C

Explanation

We have
`f(x)=tan^(-1)((x^(2)+1)/(x^(2)+sqrt3)),x in R`
`"Let "y=(x^(2)+1)/(x^(2)+sqrt3)`
`rArr" "yx^(2)+sqrt3y=x^(2)+1`
`rArr" "(sqrt3y-1)=(1-y)x^(2)`
`therefore" "x^(2)=((sqrt3y-1)/(1-y))ge0`
`"So, "(sqrt3y-1)/(y-1)le0`
`rArr" "y in[(1)/(sqrt3),1)`
`therefore" Range of "f(x)=[(pi)/(6),(pi)/(4))`