Mathematics - Functions Question with Solution | TestHub

$${\left({\sin }^{-1}x\right)}^3+{\left({\cos }^{-1}x\right)}^3={\left(\frac{\pi }{2}\right)}^3-3{\sin }^{-1}x{\cos }^{-1}x\left(\frac{\pi }{2}\right)$$

$$=\frac{{\pi }^3}{8}-\frac{3\pi }{2}{\sin }^{-1}x\left(\frac{\pi }{2}-{\sin }^{-1}x\right)$$

$$=\frac{{\pi }^3}{32}+\frac{3\pi }{2}{\left\{{\sin }^{-1}x-\frac{\pi }{4}\right\}}^2$$

The minimum value is $\frac{{\pi }^3}{32}$, and the maximum value is $\frac{7{\pi }^3}{8}$.