If range of function `f(x)=sin^(-1)x+2tan^(-1)x+x^2+4x+1` is `[p , q],` then the value of `(p+q)` is_______>

Answer

4

Explanation

`f(x) = sin^(-1) x + 2 tan^(-1) x + (x + 2)^(2) -3`
Domain of `f(x) "is " [-1, 1]`
Also `f(x)` is an increasing function in the domain. Therefore, `p=f_("min") (x) = f(_1) = -(pi)/(2) + 2 ((-pi)/(4)) + 1 -3 = - pi -2`
and `q = f_("max") (x) = f(1) = (pi)/(2) = (pi)/(2) + 2 ((pi)/(4)) + 9 -3 = pi + 6`
Therefore, the range of `f(x) "is " [-pi -2, pi + 6]`
Hence, `(p + q) = 4`