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Question

The greatest and least values of (sin1x)3+(cos1x)3 are


A

π2,π2

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B

π32,π32

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C

π332,7π38

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D

None

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Solution

The correct option is C

π332,7π38


=(sin1x+cos1x)33sin1xcos1x(sin1x+cos1x)=π383(sin1xcos1x)π2=π383π2sin1x(π2sin1x)π38+3π2[(sin1x)2π2sin1x]π38+3π2[(sin1xπ4)2]3π332=π332+3π2(sin1xπ4)2
So, the least value is π332.

(sin1xπ4)2(3π4)2
Because when x=1 sin1xπ4=π2π4=3π4
Greatest value is π332+9π216×3π2=7π38


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