Find the maximum and minimum values of `(sin^(-1)x)^3+(cos^(-1)x)^3,` where `-1lt=xlt=1.`

Explanation

`y=(sin ^{-1} x)^{3}+(cos ^{-1} x)^{3}`



`implies (sin ^{-1} x+cos ^{-1} x)^{3}-3 sin ^{-1} x cos ^{-1} x(sin ^{-1} x+cos ^{-1} x)`



`implies frac{pi^{3}}{2^{3}}-3 sin ^{-1} x cos ^{-1} x(frac{pi}{2})(because quad sin ^{-1} x+cos ^{-1} x=frac{pi}{2})`



`implies frac{pi^{3}}{8}-3 sin ^{-1} x cos ^{-1} x(frac{pi}{2})`



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