Question

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

• A. $x<-\frac{1}{2}$
• B. $x>\frac{1}{2}$
• C. $0<x<\frac{1}{2}$
• D. $-\frac{1}{2}<x<0$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $x<-\frac{1}{2}$
B $x>\frac{1}{2}$
C $0<x<\frac{1}{2}$
D $-\frac{1}{2}<x<0$

We are given $y=2x^2-\log \left|x\right|,(x\ne 0).$
Hence
$\frac{dy}{dx}=4x-\frac{1}{x}=\frac{(2x-1)(2x+1)}{x}$ for $x\ne 0.$
$\frac{dy}{dx}=\frac{4}{x^2}[x-(-\frac{1}{2})][x-0][x-\frac{1}{2}]$
Clearly
$\frac{dy}{dx}=$+ive for $x>\frac{1}{2}\Rightarrow$ increasing
-ive for $0<x<\frac{1}{2}\Rightarrow$ decreasing
+ive for $-\frac{1}{2}<x<0\Rightarrow$ increasing
-ive for $x<-\frac{1}{2}\Rightarrow$ decreasing