Least value of ${(\sin^{- 1} x)}^{3} + {(\cos^{- 1} x)}^{3}$ is

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$\frac{π^{3}}{32}$

$- \frac{π^{3}}{8}$

$- \frac{π^{3}}{32}$

$\frac{π^{3}}{8}$

answer is C.

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Detailed Solution

Least value of ${(\sin^{- 1} x)}^{3} + {(\cos^{- 1} x)}^{3}$ is

$\begin{array}{l} {(\sin^{- 1} x)}^{3} + {(\cos^{- 1} x)}^{3} = {(\sin^{- 1} x + \cos^{- 1} x)}^{3} - 3 \sin^{- 1} x \cos^{- 1} x (\sin^{- 1} x + \cos^{- 1} x) \\ = {(\frac{π}{2})}^{3} - 3 \sin^{- 1} x \cos^{- 1} x . \frac{π}{2} \\ = \frac{π^{3}}{8} - \frac{3 π}{2} \sin^{- 1} x (\frac{π}{2} - \sin^{- 1} x) \end{array}$ $\begin{array}{l} = \frac{π^{3}}{8} - \frac{3 π^{2}}{4} \sin^{- 1} x + \frac{3 π}{2} {(\sin^{- 1} x)}^{2} \\ = \frac{π^{3}}{8} + \frac{3 π}{2} {(\sin^{- 1} x - \frac{π}{4})}^{2} - \frac{3 π^{3}}{32} \\ = \frac{π^{3}}{32} + \frac{3 π}{2} {(\sin^{- 1} x - \frac{π}{4})}^{2} (∵ {(\sin^{- 1} x - \frac{π}{4})}^{2} \leq {(\frac{3 π}{4})}^{2}) \end{array}$