Find the greatest and least value of `(sin^-1x)^3+(cos^-1x)^3.`

Options

A. `(pi^(3))/32`
B. `(pi^(3))/8`
C. `(7pi^(3))/8`
D. none of these

Answer

C

Explanation

We have `(sin^(-1)x)^(3)+(cos^(-1)x)^(3)`
`=(sin^(1)x+cos^(-1)x)^(3)-3sin^(-1)xcos^(-1)x(sin^(-1)x+cos^(-1)x)`
`=(pi^(3))/8-=3(sin^(-1)xcos^(-1)x)(pi)/2`
`=(pi^(3))/8-(3pi)/2x((pi)/2-sin^(-1)x)`
`=(pi^(3))/8-(3pi^(2))/4sin^(-1)x+(3pi)/2(sin^(-1)x)^(2)`
`=(pi^(3))/8+(3pi)/2[(sin^(-1)x)^(2)-(pi)/2sin^(-1)x]`
`=(pi^(3))/8+(3pi)/2[(sin^(-1)x-(pi)/4)^(2)]-(3pi^(3))/32`
`=(pi^(3))/32+(3pi)/2(sin^(-1)x-(pi)/4)^(2)`
and since `(sin^(-1)x-(pi)/4)^(2)le(-(3pi)/4)^(2)`
`:.` The greatest value is
`(pi^(3))/32+(9pi^(2))/16xx(3pi)/2=(7pi^(3))/8`