Question

Greatest value of $(si{n}^{-1}x{)}^3+(co{s}^{-1}x{)}^3$ is :

• A. $\frac{7{\pi }^3}{32}$
• B. $\frac{{\pi }^3}{32}$
• C. $\frac{7{\pi }^3}{8}$
• D. None

Answer 1

The correct option is C $\frac{7{\pi }^3}{8}$

$(si{n}^{-1}x{)}^3+(co{s}^{-1}x{)}^3$
$=(si{n}^{-1}x+co{s}^{-1}x{)}^3-3si{n}^{-1}xco{s}^{-1}x(si{n}^{-1}x+co{s}^{-1}x)$
$=\frac{{\pi }^3}{8}-\frac{3\pi }{2}si{n}^{-1}xco{s}^{-1}x$
Least value of $si{n}^{-1}x.co{s}^{-1}x=(\frac{-\pi }{2}.\pi )atx=-1$
$\therefore$ greatest value of $(si{n}^{-1}x{)}^3+(co{s}^{-1}x{)}^3$
$=\frac{{\pi }^3}{8}+\frac{3{\pi }^3}{4}=\frac{7{\pi }^3}{8}$