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Question
Find the intervals in which f(x) = log (1 + x) −\[\frac{x}{1 + x}\] is increasing or decreasing ?
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Solution
\[f\left( x \right) = \log \left( 1 + x \right) - \frac{x}{1 + x}\]
\[\text { Domain of f }\left( x \right) \text { is }\left( - 1, \infty \right).\]
\[f'\left( x \right) = \frac{1}{1 + x} - \left\{ \frac{1 + x - x}{\left( 1 + x \right)^2} \right\}\]
\[ = \frac{1}{1 + x} - \frac{1}{\left( 1 + x \right)^2}\]
\[ = \frac{x}{\left( 1 + x \right)^2}\]
\[\text { For }f(x) \text { to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow \frac{x}{\left( 1 + x \right)^2} > 0\]
\[ \Rightarrow x > 0 \left[ \because \left( 1 + x \right)^2 >0, \text { Domain }:\left( - 1, \infty \right) \right]\]
\[ \Rightarrow x \in \left( 0, \infty \right)\]
\[\text { So, f(x) is increasing on } \left( 0, \infty \right) . \]
\[\text { Forf(x) to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow \frac{x}{\left( 1 + x \right)^2} < 0\]
\[ \Rightarrow x < 0 \left[ \because \left( 1 + x \right)^2 >0, \text{Domain }:\left( - 1, \infty \right) \right]\]
\[ \Rightarrow x \in \left( - 1, 0 \right)\]
\[\text { So,f(x)is decreasing on }\left( - 1, 0 \right).\]
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