Find the value of x for which `f(x) = 2 sin^(-1) sqrt(1 - x) + sin^(-1) (2 sqrt(x - x^(2)))` is constant

Explanation

`f(x) = 2 sin^(-1) sqrt(1 - (sqrtx)^(2)) + sin^(-1) (2 sqrt((x)^(2) {1 - (sqrtx)^(2)})`
Put `sqrtx = cos theta, " where " theta = cos^(-1) sqrtx in [0, (pi)/(2)]`
`:. F(x) = 2 sin^(-1) sqrt(1 - cos^(2) theta) + sin^(-1) 2sqrt((cos^(2) theta) (1 - cos^(2) theta))`
`= 2 sin^(-1) (sin theta) + sin^(-1) (2 sin theta cos theta)`
`= 2 sin^(-1) (sin theta) + sin^(-1) (sin 2 theta)`
`= 2 theta + sin^(-1) (2 theta)`
`= 2 theta + (pi - 2 theta) " if " 2 theta in [(pi)/(2), pi]`
`= pi, theta in [(pi)/(4), (pi)/(2)]`
So, `sqrtx = cos theta in [0, (1)/(sqrt2)]`
or `x in [0, (1)/(2)]`