If f(x) = co s ^ - 1 (x(3)/(2) - 1 - x - x ^ 2 + x ^ 3 ), 0 ≤ x ≤ 1...

$f\left(x\right) = co{s}^{- 1}\left\{x\sqrt{x} - \sqrt{\left(1 - x\right)\left(1 - {x}^{2}\right)}\right\}$
$= co{s}^{- 1}\left\{x\sqrt{x} - \sqrt{1 - {\left(\sqrt{x}\right)}^{2}}\sqrt{1 - {x}^{2}}\right\}$
$= co{s}^{- 1}x + co{s}^{- 1}\sqrt{x}$

At $x = 1$,

minimum value of $f ( x ) = 0$