`if y=tan^(-1)x+"tan"^(-1)(1)/(x)+sec^(-1)x,then" "yin`

Options

A. `[(pi)/(2),pi)uu(pi,(3pi)/(2)]`
B. `[(pi)/(2),(3pi)/(2)]`
C. `(o,pi)`
D. `[o,(pi)/(2))uu[(pi)/(2),pi)`

Answer

3

Explanation

`therefore` Domain is x`in(-oo,-1]cup[1,oo)`
case(1):`x ge1`,
`y=tan^(-1)x+ tan^(-1)""(1)/(x)+sec^(-1)x`
`=(pi)/(2)+sec^(-1)xforall sec^(-1)x in[0,(pi)/(2))y in[(pi)/(2),pi)`
case(2), `xle-1,y=tan^(-1)x+tan^(-1)""(1)/(x)+sec^(-1)x`
`tan^(-1)x+cot^(-1)x-pi+sec^(-1)x`
`=sec^(-1)-(pi)/(2) forallsec^(-1)x in((pi)/(2),pi]`
so `y in (o,(pi)/(2)]`
`therefore y in (o,pi)`