The function `f(x)=2|x|+|x+2|=||x|2|-2|x||` has a local minimum or a local maximum at `x=` `-2` (b) `-2/3` (c) 2 (d) `2/3`

Options

A. `-2 and -2/3 `
B. (b)-2 and 0`
C. `-2/3 and 2 `
D. `2 and -2`

Answer

A

Explanation

We have ,
f(x)=2|x|+|x-2|-||x+2|-2|x|
`rArr`
`f(x)={{:(-2x-(x+2)-|x-2|=-2x-x-2+x-2=-2x-4 "," x lt -2 ),(-2x+x+2-|x+2+2x|=-2x+x+2+3x+2=2x+4","-2 le x lt -2/3),(-2x+x+2-|x+2+2x|=-2x+x+2-3x-2=-4x","-2/3lt x lt 0 ),(2x+x+2-|x+2-2x|=2x+x+2-(2-x)=4x"," 0 le x le 2),(2x+x+2-|x+2-2x|=2x+x+2-x=2x+4 "," xgt 2 ):}`
The graph of f(x) is shown in the following figure.
Clearly,x=-2 ,0 are points of local minima and `x=-2/3` is a points of local maximum .