Question

Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when ______.

Options

• x < 0

• x > 0

• x ∈ R

• x > –1

Answer 1

Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when x > –1.

Explanation:

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

Differentiating w.r.t.x, we get

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

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

f'(x) = `1/(1 + x) - 4/(2 + x)^2 > 0`

⇒ `1/(1 + x) > 4/(2 + x)^2`

⇒ (2 + x)² > 4(1 + x)

⇒ 4 + x² + 4x > 4 + 4x

⇒ x² > 0

and f'(0) = 0

So, for all x > –1, f(x) increases