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Question
The function \[f\left( x \right) = \frac{x}{1 + \left| x \right|}\] is
Options
strictly increasing
strictly decreasing
neither increasing nor decreasing
none of these
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Solution
strictly increasing
\[f\left( x \right) = \frac{x}{1 + \left| x \right|}\]
\[\text { Case 1: When }x > 0, \left| x \right| = x\]
\[f\left( x \right) = \frac{x}{1 + \left| x \right|}\]
\[ = \frac{x}{1 + x}\]
\[ \Rightarrow f'\left( x \right) = \frac{\left( 1 + x \right)1 - x\left( 1 \right)}{\left( 1 + x \right)^2}\]
\[ = \frac{1}{\left( 1 + x \right)^2} > 0, \forall x \in R\]
\[\text { So,f }\left( x \right) \text { is strictly increasing when }x> 0.\]
\[\text { Case 2: When }x < 0, \left| x \right| = - x\]
\[f\left( x \right) = \frac{x}{1 + \left| x \right|}\]
\[ = \frac{x}{1 - x}\]
\[ \Rightarrow f'\left( x \right) = \frac{\left( 1 - x \right)1 - x\left( - 1 \right)}{\left( 1 - x \right)^2}\]
\[ = \frac{1}{\left( 1 - x \right)^2} > 0, \forall x \in R\]
\[\text { So,f }\left( x \right) \text { is strictly increasing when }x <0.\]
\[\text { Thus,f }\left( x \right) \text { is strictly increasing on R } . \]
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