Question

The function `f(x) = log(1 + x) - (2x)/(2 + x)` is increasing on

Options

• `(0, oo)`

• `(- oo, 0)`

• `(- oo, oo)`

• None of these

Answer 1

`(0, oo)`

Explanation:

Given `f(x) = log(1 + x) - (2x)/(2 + x)`

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

= `1/(1 + x) - 4/(2 + x)^2`

= `((2 + x)^2 - 4 - 4x)/((1 + x)(2 + x)^2`

= `x^2/((1 + x)(2 + x)^2` > 0 for all x ∈ (0, `oo`)

Thus, given function f(x) is increasing on (0, `oo`).