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

Answer

`(pi^(2))/(16)`

Explanation

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)`