Question

Find the intervals of monotocity of the function $f\left(x\right)=2x^2-\log |x|,x\ne 0$.

Answer 1

We have $f\left(x\right)=2x^2-\log |x|,x\ne 0$

Now, $f^{\prime }\left(x\right)=4x-\frac{1}{x}=\frac{4x^2-1}{x}$

For $f\left(x\right)$ being monotone increasing we must have

$f^{\prime }\left(x\right)\ge 0$

or, $4x^2-1\ge 0$

or, $x\in (-\infty ,-\frac{1}{2}]\cup [\frac{1}{2},\infty )$

For $f\left(x\right)$ being monotone decreasing we must have

$f^{\prime }\left(x\right)\le 0$

or, $4x^2-1\le 0$

or, $x\in [-\frac{1}{2},\frac{1}{2}]-\left\{0\right\}$