Range of `sin^(-1)((x^(2)+1)/(x^(2)+2))` is

Options

A. `[0,(pi)/(2)]`
B. `(0,(pi)/(6))`
C. `[(pi)/(6),(pi)/(2))`
D. None of these

Answer

C

Explanation

Here, `(x^(2)+1)/(x^(2)+2)=1-(1)/(x^(2)+2)`
Now, `2 le x^(2)+2lt oo` for all x ein R
`implies(1)/(2)ge (1)/(x^(2)+2)gt0`
`implies -(1)/(2) le (-1)/(x^(2)+2) lt 0`
`implies(1)/(2) le 1-(1)/(x^(2)+2)lt`
`implies (pi)/(6) le sin^(-1)(1-(1)/(x^(2)+2))lt(pi)/(2)`