Question

If $a<\frac{1}{32}$, then the number of solutions of ${\left({\sin }^{-1}x\right)}^3+{\left({\cos }^{-1}x\right)}^3=a{\pi }^3$, is

• A. $0$
• B. $1$
• C. $2$
• D. infinite

Answer 1

The correct option is A $0$

$f\left(x\right)=\left(si{n}^{-1}\right(x){)}^3+(co{s}^{-1}\left(x\right){)}^3$
The least value of the function comes at $x=\frac{1}{\sqrt{2}}$ and greatest value come for $x=-1$.
$f\left(\frac{1}{\sqrt{2}}\right)=(\frac{\pi }{4}{)}^3+(\frac{\pi }{4}{)}^3$
$=\frac{2{\pi }^3}{64}$
$=\frac{{\pi }^3}{32}$
And $f(-1)=-\frac{{\pi }^3}{8}+{\pi }^3$
$=\frac{7{\pi }^3}{8}$
Therefore $aϵ[\frac{1}{32},\frac{7}{8}]$
Hence for $a<\frac{1}{32}$ $f\left(x\right)$ has no solution.