Mathematics - Application of Derivative Question with Solution | TestHub

$f\left(x\right)=\frac{1}{x}{\log }_e\left(1+x\right), x\ne 0$

$f^{\text{'}}\left(x\right)=\frac{\frac{x}{1+x}-\ln \left(1+x\right)}{x^2}=\frac{x-\left(1+x\right)\ln \left(1+x\right)}{x^2\left(1+x\right)}$

Let $g\left(x\right)=x-\left(1+x\right)\ln \left(1+x\right)$

$g^{\text{'}}\left(x\right)=-\ln \left(1+x\right)$

$\Rightarrow g^{\text{'}}\left(x\right)\left\{\begin{array}{ll}>0& \forall x\in \left(-1,0\right)\\ <0& \forall x\in \left(0,\infty \right)\end{array}\right.$

$g_{max}$ at $x=0\Rightarrow g\left(0\right)=0$ is the maximum value

So, $g\left(x\right)<0\forall x\in \left(-1,\infty \right) \Rightarrow f^{\text{'}}\left(x\right)<0\forall x\in \left(-1,\infty \right)$

Hence, $f\left(x\right)$ is decreasing $\forall x\in \left(-1,\infty \right)$