Question

Find the intervals of monotonicity of the function

$\frac{x}{\log x}$

• A. Increasing in $(0,e)$
• B. Increasing in $(e,\infty )$
• C. decreasing in $(0,e)-\left\{1\right\}$
• D. Decreasing in $(e,\infty )$

Answer 1

Answer: (B), (C)

The correct options are
B Increasing in $(e,\infty )$
C decreasing in $(0,e)-\left\{1\right\}$

Let $f\left(x\right)=\frac{x}{\log x}$ $\therefore$ Domain of f is $(0,\infty )-\left\{1\right\}$
Now $f^{\prime }\left(x\right)=\frac{\log x-1}{(\log x{)}^2}$
For $f$ to be increasing $f^{\prime }\left(x\right)>0\Rightarrow \log x-1>0\Rightarrow x>e$
and for $f$ to be decresing, $f^{\prime }\left(x\right)<0\Rightarrow \log x-1<0\Rightarrow x<e$ also $x>0\ne 1\Rightarrow xϵ(0,e)-\left\{1\right\}$