Question

The function $y=\frac{x}{\ln x}$ increases in the interval-

• A. $(-\infty ,0)$
• B. $(e,\infty )$
• C. $(0,\infty )$
• D. $(-\infty ,e)$

Answer 1

Answer: (B)

$(e,\infty )$

$f\left(x\right)=\frac{x}{\log x}$
$f^{\prime }\left(x\right)=\frac{\log x-x\frac{1}{x}}{(\log x{)}^2}$
For $f\left(x\right)$ to be increasing,
$f^{\prime }\left(x\right)>0$
$\Rightarrow \frac{\log x-1}{(\log x{)}^2}>0$
$\Rightarrow \log x>1$
$\Rightarrow x>e$
$\Rightarrow x\in (e,\infty )$