Question

$f\left(x\right)=\frac{x}{\log x}-\frac{\log 5}{5}$ is decreasing in

• A. $(e,\infty )$
• B. $(0,1)U(1,e)$
• C. (0, 1)
• D. (1, e)

Answer 1

Answer: (B)

$(0,1)U(1,e)$

$f\left(x\right)=\frac{x}{\log x}-\frac{\log 5}{5}$ is decreasing in ....

For decreasing functions

$f^{\prime }\left(x\right)<0$

$\frac{\log x\times \frac{d}{dx}\left(x\right)-x\times \frac{d}{dx}\left(\log x\right)}{{\left(\log x\right)}^2}-0<0$

$\frac{\log x-1}{{\left(\log x\right)}^2}<0$

${\log }_{e}x-1<0$

${\log }_{e}x-1<1$

$x<e$

And also $x>0$

Now, at $x=1,f^{\prime }\left(x\right)$ wont exit

Therefore, $(0,1)\bigcup (1,e)$ is the correct answer.