Question:
Mark the correct alternative in the following:
The function $f(x)=\frac{x}{1+|x|}$ is
A. strictly increasing
B. strictly decreasing
C. neither increasing nor decreasing
D. none of these
Solution:
Formula:- (i) The necessary and sufficient condition for differentiable function defined on (a,b) to be strictly increasing on $(a, b)$ is that $f^{\prime}(x)>0$ for all $x \in(a, b)$
$f(x)=\frac{x}{1+|x|}$
For $x>0$
$\frac{\mathrm{d}(\mathrm{f}(\mathrm{x}))}{\mathrm{dx}}=\frac{1}{1+\mathrm{x}^{2}}=\mathrm{f}^{\prime}(\mathrm{x})$
For $x<0$
$\frac{\mathrm{d}(\mathrm{f}(\mathrm{x}))}{\mathrm{dx}}=\frac{1}{1-\mathrm{x}^{2}}=\mathrm{f}^{\prime}(\mathrm{x})$
Both are increasing for $f^{\prime}(x)>0$