Question

The function $f(x) =\frac {x} {1+|x|}$ is

• A. strictly increasing
• B. strictly decreasing
• C. neither increasing nor decreasing
• D. not differentiable at x = 0.

Answer 1

Answer: (A)

$f'\left(x\right) = \frac{\left(1+\left|x\right|\right)\cdot1-x\left(\pm1\right)}{\left[1+\left|x\right|\right]^{2}}$
$= \frac{1+\left|x\right|-\left|x\right|}{\left(1+\left|x\right|\right)^{2}}$
$= \frac{1}{\left(1+\left|x\right|\right)^{2}} > 0 \,\forall\,x \in R$
$\therefore f$ is strictly increasing.