Question

Find the interval of increase and decrease of the following functions.
$f\left(x\right)=\frac{x}{\ln x}$

Answer 1

$f\left(x\right)=\frac{x}{\ln \left(x\right)}$

First, $f^{\prime }\left(x\right)=\frac{\ln \left(x\right)-\frac{x}{x}}{(\ln x{)}^2}$

$\Rightarrow \frac{\ln x-1}{(\ln x{)}^2}=0$

$\Rightarrow \ln =1$

$\Rightarrow x=e$

Domain of $f\left(x\right):0<x<1$ or $x>1$

because $\ln 0$ is not defined and $\ln 1=0$.

The function monotone intervals are : -

$0<x<1,1<x<e,e<x<\infty$

At, $0<x<1$,

$f^{\prime }\left(x\right)=-\frac{\ln x-1}{{\ln }^{2}x}<0$, therefore decreasing

At, $1<x<e$,

$f^{\prime }\left(x\right)=\frac{\ln x-1}{{\ln }^{2}x}<0$, therefore decreasing

At, $e<x<\infty$,

$f^{\prime }\left(x\right)=\frac{\ln x-1}{{\ln }^{2}x}>0$, therefore increasing.