Find the minimum value of the function `f(x)=(pi^2)/(16cot^(-1)(-x))-cot^(-1)x`

Explanation

`f(x) = pi^2/(16cot^-1(-x)) - cot^-1x`

`=>f(x) = pi^2/(16cot^-1(-x)) - (pi-cot^-1(-x))`

`=>f(x) = cot^-1(-x)+pi^2/(16cot^-1(-x)) - pi`

`=>f(x) = (sqrt(cot^-1(-x)) - pi/(4sqrt(cot^-1(-x))))^2+pi/2 - pi`

Now, minimum value of `(sqrt(cot^-1(-x)) - pi/(4sqrt(cot^-1(-x))))^2` will be `0` as square of a number can not be less than `0`.

`:. f(x)_min = pi/2 - pi = -pi/2.`