The range of `f(x)=sin^(-1)((x^2+1)/(x^2+2))` is `(a)[0,pi/2]` (b) `(0,pi/6)` (c) `[pi/6,pi/2]` (d) none of these

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)`