On which of the following intervals is the funciotn  $f\left(x\right)=2x^2-\log \left|x\right|,x\ne 0$  increasing ?

$f\left(x\right)=2x^2-\log \left|x\right|=\{\begin{array}{l}2x^2-\log x,x>0\\ 2x^2-\log (-x),x<0\end{array}$

$\Rightarrow f^{\mathrm{\prime }}\left(x\right)=4x-\frac{1}{x},\mathrm{\forall }x\ne 0$

For f(x) to be increasing,  $f^{\mathrm{\prime }}\left(x\right)>0$

$\Rightarrow \frac{(2x-1)(2x+1)}{x}>0\Rightarrow x\in (-\frac{1}{2},0)\cup (\frac{1}{2},\mathrm{\infty })$