If `cos^(-1)(2-x^(2))+sin^(-1)(2-x^(2))+tan^(-1)xge(2pi)/3` have no solution if `xltalpha` or `xgt beta`(for `alphalt beta`) then `{1/("max"(alpha))+("min"(beta))^(2)}=`………

Answer

4

Explanation

`:'tan^(-1)xge(2pi)/3-(pi)/2` of `=1le2-x^(2)le1`
`impliestan^(-1)xge(pi)/6 -3ge-x^(2)ge-1`
`x ge 1/(sqrt(3)) 1 le |x| le sqrt(3)`
`:.1le x le sqrt(3)`
`:.` have no solution for `xlt1 `or `xgt sqrt(3)`
So, `alpha_("max")=1` and `beta_("min")=sqrt(3)`