Question

Let a function $f\left(x\right)=x^2-4|x|,$ then which of the following is correct

• A. $f\left(x\right)$ is strictly increasing in $(-\infty ,-2)$
• B. $f\left(x\right)$ is strictly increasing in $(-1,1)$
• C. $f\left(x\right)$ is strictly decreasing in $(-2,2)$
• D. $f\left(x\right)$ is strictly decreasing in $(1,2)$

Answer 1

The correct option is D $f\left(x\right)$ is strictly decreasing in $(1,2)$

Given : $f\left(x\right)=x^2-4|x|$
As, $f(-x)=f\left(x\right);$ function is even(symmetric about $Y-$axis)

For $x>0,f\left(x\right)=x^2-4x$
$\Rightarrow f^{\prime }\left(x\right)=2x-4=0,$ gives $x=2$
$\Rightarrow f^{\prime }\left(x\right)<0,$ when $x\in (0,2)\to$ decreasing.
$f^{\prime }\left(x\right)>0,$ when $x\in (2,\infty )\to$ increasing.

Now for $x<0;f\left(x\right)=x^2+4x$
$\Rightarrow f^{\prime }\left(x\right)=2x+4=0,$ gives $x=-2$
$\Rightarrow f^{\prime }\left(x\right)<0,$ when $x\in (-\infty ,-2)\to$ decreasing.
$f\left(x\right)>0,$ when $x\in (-2,0)\to$ increasing.