The maximum value of `f(x)=tan^(-1)(((sqrt(12)-2)x^2)/(x^4+2x^2+3))` is (A) `18^@` (B) `36^@` (C) `22.5^@` (D) `15^@`

Options

A. `18^(@)`
B. `36^(@)`
C. `22.5^(@)`
D. `15^(@)`

Answer

D

Explanation

`f(x) = tan^(-1). (((sqrt12 - 2)x^(2))/(x^(4) + 2x^(2) + 3))`
`= tan^(-1) ((2(sqrt3 -1))/(x^(2) + (3)/(x^(2)) + 2))`
As `x^(2) + (3)/(x^(2)) ge 2 sqrt3` [using A.M. `ge` G.M.]
`rArr x^(2) + (3)/(x^(2)) + 2 ge 2 + 2 sqrt3`
`:. (f(x))_("max") = tan^(-1) ((2(sqrt3 -1))/(2(sqrt3 + 1)))`
`= (pi)/(12)`