Find the range of f(X) =`tan^(-1)x-(1)/(2)log_(e)x in ((1)/(sqrt(3)),sqrt(3))`

Answer

`[(pi)/(3)-log_(e)sqrt(3),(pi)/(6)-log_(e)(1)/sqrt(3)]`

Explanation

`f(x)= tan^(-1)x-log_(e)x`
`therefore f(x)=(1)/(1+x^(2))-(1)/(x)=(x^(2)+1-x)/(x^(2)+1) lt 0 forall x in [(1)/sqrt(3),sqrt(3)]`
So `f_(min)=f(sqrt(3))=tan^(-1)sqrt(3)-log_(e)sqrt(3)=(pi)/(3)-log_(e)sqrt(3)`
`f_(max)=f(1)/(sqrt(3))=tan^(-1)(1)sqrt(3)-log_(e)sqrt(3)=(pi)/(6)-log_(e)(12)/(sqrt(3))`
Hence range is `[(pi)/(3)-log_(e)sqrt(3),(pi)/(6)-log_(e)(1)/(sqrt(3))]`