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Find the greatest and least value of `(sin^-1x)^3+(cos^-1x)^3.`A. `(pi^(3))/32`B. `(pi^(3))/8`C. `(7pi^(3))/8`D. none of these |
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Answer» Correct Answer - C 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` |
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