Range of `f(x)=sin^(-1)x+cos^(-1)x+cot^(-1)x`

Answer

`[5-(5pi)/(4),9+(pi)/(4)]`

Explanation

f(x)=`sin^(-1)x-cot ^(-1)x+x^(2)+2x+6`
Domain of the f(x) is [-1,1]
Now f'(x) =`(1)/sqrt(1-x^(2))+(1)/(1+x^(2))+2x+2`
For `x in [-1,1],2x+2gt0`
`therefore f(x)gt0`
So f(x) is an increasing funciton
`therefore f_(min)=f(-1) = (pi)/(2)-(3pi)/(4)+5=5-(5pi)/(4)`
`f_(max)=f(x1)=(pi)/(2)=(pi)/(4)+9=9+(pi)/(4)`
`therefore` Range is `[5-(5pi)/(4),9+(pi)/(4)]`