Question

The greatest value of the function f(x) = `tan^-1x - 1/2logx` in `[1/sqrt(3), sqrt(3)]` is ______.

Options

• `π/6 + 1/4log3`

• `π/3 - 1/4log3`

• `π/6 - 1/4log3`

• `π/3 + 1/4log3`

Answer 1

The greatest value of the function f(x) = `tan^-1x - 1/2logx` in `[1/sqrt(3), sqrt(3)]` is `underlinebb(π/6 + 1/4log3)`.

Explanation:

f'(x) = `1/(1 + x^2) - 1/(2x)`

= `(-(x - 1)^2)/(2x(x^2 + 1))`

f(x) in (0, ∞)

∴ f(x)_max = `f(1/sqrt(3))`

`f(1/sqrt(3)) = π/6 - 1/2log  1/sqrt(3)`

= `π/6 + 1/4log3`