Question

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

• A. $(-\frac{1}{2},\frac{1}{2})$
• B. ($-\infty ,-\frac{1}{2})$
• C. $(-\infty ,-\frac{1}{2})\cup (0,\frac{1}{2})$
• D. $(-\infty ,-\frac{1}{2})\cup (\frac{1}{2},\infty )$

Answer 1

The correct option is C $(-\infty ,-\frac{1}{2})\cup (0,\frac{1}{2})$

$f\left(x\right)=2x^2-\log |x|$ ($x\ne 0$)
$f\left(x\right)=2x^2-\log x$
$f^{\prime }\left(x\right)=4x-\frac{1}{x}$
For decreasing function,
$f^{\prime }\left(x\right)<0$
$\frac{4x^2-1}{x}<0$
$\frac{(2x+1)(2x-1)}{x}<0$
$\Rightarrow x\in (-\infty ,-\frac{1}{2})\cup (0,\frac{1}{2})$