If x>0 then the interval of monotonically increasing of $\mathrm{f}\left(\mathrm{x}\right)=2{\mathrm{x}}^2-\mathrm{logx}$  is

$\begin{array}{l}\mathrm{Given} \mathrm{f}\left(\mathrm{x}\right)=2{\mathrm{x}}^2-\mathrm{logx}\\ diff w.r.to x on both sides\\ {\mathrm{f}}^{\text{'}}\left(\mathrm{x}\right)=4\mathrm{x}-\frac{1}{\mathrm{x}}\\ \sin ce f\left(x\right) is monotonically increa\sin g then f\text{'}\left(x\right)\ge 0\\ \\ \Rightarrow \frac{4{\mathrm{x}}^2-1}{\mathrm{x}}>0(\because x>0)\\ \frac{(2\mathrm{x}-1)(2\mathrm{x}+1)}{\mathrm{x}}>0\\ \Rightarrow (x-(-\frac{1}{2}\left)\right)(x-\frac{1}{2})>0\\ \Rightarrow x\in (-\infty ,-\frac{1}{2})\cup (\frac{1}{2},\infty )\\ \Rightarrow x\in (\frac{1}{2},\infty )(\because x>0)\end{array}$