Question

On which of the following intervals is the function $f\left(x\right)=2x^2-\log |x|,x\ne 0$ increasing in

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

Answer 1

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

$f\left(x\right)=2x^2-\log |x|$
$=\{\begin{array}{cc}2x^2-\log x& x>0\\ 2x^2-\log (-x)& x<0\end{array}$
$\Rightarrow f^{\prime }\left(x\right)=4x-\frac{1}{x},\forall x\ne 0$
For $f\left(x\right)$ to be increasing
$f^{\prime }\left(x\right)>0\Rightarrow 4x-\frac{1}{x}>0$
$\Rightarrow \frac{4x^2-1}{x}>0$
$\Rightarrow \frac{(2x-1)(2x+1)}{x}>0$
$\Rightarrow x(2x-1)(2x+1)>0$
$\Rightarrow xϵ(-\frac{1}{2},0)\cup (\frac{1}{2},\infty )$.