The number of integral values in the range of the function `f(x)=sin^(-1)x-cot^(-1)x+x^(2)+2x +6` is

Options

A. 10
B. 11
C. 12
D. 8

Answer

D

Explanation

`f(x)=sin^(-1)x-((pi)/(2)-tan^(-1)x)+(x+1)^(2)+5`
`=sin^(-1)x + tan^(-1)x-(pi)/(2)+(x +1)^(2)+5`
Domain of f(x) is [-1,1] and `sin^(-1) x, tan^(-1)x` and `(x+1)^(2)` are all increasing in [-1,1]
`therefore f(x)` is increasing
`f(-1)=(-5pi)/(4)+5` and `f(1)=(pi)/(4)+9`
`therefore` Range of f(x) is `[(-5pi)/(4)+5,(pi)/(4)+9]`