If `alpha` is an integer satisfying `|alpha|lt=4-|[x]|,` where `x` is a real number for which `2xtan^(-1)x` is greater than or equal to `ln(1+x^2),` then the number of maximum possible values of `a` (where [.] represents the greatest integer function) is_____

Answer

9

Explanation

(9) Let y =2x `tan^(-1)x-ln(1+x^(2))`
`y=2 tan^(-1)x+(2x)/(1+x^(2))-(2x)/(1+x^(2))`
`therefore y gt 0 forall lx in R^(+) ,y lt0 forall x in R^(-)`
Therefore 4-|[x]| takes the values 0,1,2,3,4
`|alpha|le4-|[x]|` is satisfied by`alpha=0 pm1,pm 2,pm 3,pm 4`
Therefore number of values of `alpha` is 9