Question

The maximum value of $f\left(x\right)=\frac{\log x}{x}(x\ne 0,x\ne 1)$ is

• A. $e$
• B. $\frac{1}{e}$
• C. $e^2$
• D. $\frac{1}{e^2}$

Answer 1

The correct option is B $\frac{1}{e}$

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

$f^{\prime }\left(x\right)=\frac{x.\frac{1}{x}-\log x}{x^2}=\frac{1-\log x}{x^2}$

$f^{\prime }\left(x\right)=0\Rightarrow \frac{1-\log x}{x^2}=0$

$⟹\log x=1$

$⟹x=e$

max. value $f\left(x\right)=f\left(e\right)=\frac{\log e}{e}=\frac{1}{e}$