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Find the maximum value of (sec^(-1) x) (...

Find the maximum value of `(sec^(-1) x) (cosec^(-1) x), x ge 1`

Text Solution

Verified by Experts

The correct Answer is:
`(pi^(2))/(16)`
For `x ge 1, cosec^(-1) x gt 0`
Using A.M. `ge` G.M., we have
`(sec^(-1) x + cosec^(-1) x)/(2) ge sqrt(sec^(-1) x cosec^(-1) x)`
`rArr ((pi//2))/(2) ge sqrt(sec^(-1) x cosec^(-1) x)`
`rArr (pi)/(2) ge sqrt(sec^(-1) x cosec^(-1) x)`
`:. sec^(-1) x cosec^(-1) x le (pi^(2))/(16)`
Thus maximum value of `sec^(-1) x cosec^(-1) x " is " (pi^(2))/(16)`
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