Question

$f\left(x\right)=\frac{x}{{\log }_{x}e};x\in R^{+}-\left\{1\right\}$ is increasing on the interval ______

• A. $(-e,\infty )$
• B. $(\frac{1}{e},1)\cup (1,\infty )$
• C. $(0,\infty )-\left\{1\right\}$
• D. $(\frac{1}{e},\infty )$

Answer 1

The correct option is B $(\frac{1}{e},1)\cup (1,\infty )$

$f\left(x\right)=\frac{x}{{\log }_{x}e}=x{\log }_{e}x$
$\Rightarrow f^{\prime }\left(x\right)=x\times \frac{1}{x}+{\log }_{e}x\times 1$
$\Rightarrow f^{\prime }\left(x\right)=1+{\log }_{e}x$

For $f\left(x\right)$ to be increasing, $f^{\prime }\left(x\right)\ge 0$
$\Rightarrow 1+{\log }_{e}x\ge 0$
$\Rightarrow x\ge \frac{1}{e}$
Hence, the required interval is
$(\frac{1}{e},1)\cup (1,\infty )$