Question

Find the intervals in which f(x) = log (1 + x) −\[\frac{x}{1 + x}\] is increasing or decreasing ?

Answer 1

\[f\left( x \right) = \log \left( 1 + x \right) - \frac{x}{1 + x}\]

\[\text { Domain of f }\left( x \right) \text { is }\left( - 1, \infty \right).\]

\[f'\left( x \right) = \frac{1}{1 + x} - \left\{ \frac{1 + x - x}{\left( 1 + x \right)^2} \right\}\]

\[ = \frac{1}{1 + x} - \frac{1}{\left( 1 + x \right)^2}\]

\[ = \frac{x}{\left( 1 + x \right)^2}\]

\[\text { For }f(x) \text { to be increasing, we must have }\]

\[f'\left( x \right) > 0\]

\[ \Rightarrow \frac{x}{\left( 1 + x \right)^2} > 0\]

\[ \Rightarrow x > 0 \left[ \because \left( 1 + x \right)^2 >0, \text { Domain }:\left( - 1, \infty \right) \right]\]

\[ \Rightarrow x \in \left( 0, \infty \right)\]

\[\text { So, f(x) is increasing on } \left( 0, \infty \right) . \]

\[\text { Forf(x) to be decreasing, we must have }\]

\[f'\left( x \right) < 0\]

\[ \Rightarrow \frac{x}{\left( 1 + x \right)^2} < 0\]

\[ \Rightarrow x < 0 \left[ \because \left( 1 + x \right)^2 >0, \text{Domain }:\left( - 1, \infty \right) \right]\]

\[ \Rightarrow x \in \left( - 1, 0 \right)\]

\[\text { So,f(x)is decreasing on }\left( - 1, 0 \right).\]