Question

The function $f\left(x\right)=2|x|+|x+2|-\parallel x+2|-2|x\parallel$ 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)

The correct options are
A $\frac{-2}{3}$
B $-2$

$y=2|x|+|x+2|-\parallel x+2|-2|x\parallel$
Case-1 :- $x<-2$
$y=-2x-x-2-|-x-2+2x|$
$y=-2x-4$...............(1)
Case-2 :- $-2<x<0$
$y=-x+2-|3x+2|$
So, for $x<\frac{-2}{3}$
$y=2x+4$..................(2)
And for $xϵ[\frac{-2}{3},0]$
$y=-4x$..........(3)
Case 3:- $x>0$
$y=3x+2-|x-2|$
For $0<x<2$,
$y=4x$...............(4)
And for $x>2$,
$y=2x+4$............(5)
After drawing this graph, we can conclude $x=-2,0$ are point of minimum and $x=\frac{-2}{3}$ is a point of maxima