The function $y=2 x^{2}-\ln |x|$ is monotonically increasing for ____ and monotonically decreasing f

Question

The function $y=2 x^{2}-\ln |x|$ is monotonically increasing for ____ and monotonically decreasing for ____ respectively.

Professor PC Thomas Classes · Answer

$y=2 x^{2}-\ln |x|$\ $\frac{d y}{d x}=4 x-\frac{1}{x}=\frac{(2 x+1)(2 x-1)}{x}$\ Critical points are $1 / 2,-1 / 2$\ Clearly $f(x)$ is increasing on\ $(-1 / 2,0) \cup(1 / 2, \infty)$ and $f(x)$ is decreasing on $(-\infty,-1 / 2) \cup(0,1 / 2)$.\