The maximum of $f\left(x\right)=\frac{\log x}{x^2}(x>0)$  occurs at  $x=$

$f\left(x\right)=\frac{\log x}{x^2}(x>0)$
$$\begin{gathered} f\text{'}\left(x\right)=\frac{x^2{\displaystyle \frac{1}{x}}-2x\log x}{x^4} \\ =\frac{x(1-2\log x)}{x^4} \end{gathered}$$
A function $f\left(x\right)$  is said to be maximum or minimum 

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