Question

In which interval $f\left(x\right)=2x^2-\ln \left|x\right|,(x\ne 0)$ is monotonically decreasing:
(1) $(-1/2,1/2)$
(2) $(-\infty ,-1/2)$

Answer 1

$f\left(x\right)=2x^2-\ln \mid x\mid$

$f^1\left(x\right)=4x-\frac{1}{x}<0$

$⟹x(x-\frac{1}{2})(x+\frac{1}{2})<0⟹x\in (-\infty ,-\frac{1}{2})\cup (0,\frac{1}{2})$

In $\left(2\right),f\left(x\right)$ is monotonically decreasing