Question

The function $f(x)=\frac{x}{\log x}$ increases on the interval

• A. $(0,\infty )$
• B. $(0,e)$
• C. $(e,\infty )$
• D. none of these.

Answer 1

Answer: (C)

Clearly $f(x)$ is defined for $x>0$
$f^{\prime }\left(x\right)=\frac{logx\cdot 1\cdot x\cdot \frac{1}{x}}{{\left(logx\right)}^2}=\frac{logx-1}{{\left(logx\right)}^2}$
$\therefore f^{\prime }\left(x\right)>0$
$\Rightarrow logx-1>0$
$\Rightarrow logx>1$
$\Rightarrow x>e$
$\therefore x\in \left(e,\infty \right)$.