Question

The function $f\left(x\right)=2|x|+|x+2|-||x+2|-2|x||$ has a local minimum or a local maximum at $x=$

• A. $-2$
• B. $\frac{-2}{3}$
• C. $2$
• D. $\frac{2}{3}$

Answer 1

Answer: (A), (B)

$\frac{-2}{3}$

The given function is expressed as $f\left(x\right)=2|x|+|x+2|-||x+2|-2|x||$
The interval for given function can be written as,
$f\left(x\right)=-2x-4$, $x<-2$
$f\left(x\right)=2x+4$, $-2\le x\le \frac{-2}{3}$
$f\left(x\right)=-4x$, $\frac{-2}{3}\le x\le 0$
$f\left(x\right)=4x$, $0\le x<2$
$f\left(x\right)=2x+4$, $x\ge 2$


From the above graph, the points of local minima are at $x=-2$ and $x=0$. the point of local maximum is at $x=\frac{-2}{3}$.