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Question

The function f(x)=2|x|+|x+2|−||x+2|−2|x||
has a local minimum or a local maximum at x=


A

-2

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B

23

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C

2

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D

23

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Solution

The correct options are
A

-2


B

23


For f(x)=2|x|+|x+2|||x+2|2|x||
the critical points can be obtained by solving |x|=0,
|x+2|=0 and ||x+2|2|x||=0
we get x=0,-2,2, 23
Then we can write
f(x)⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ -2x-4, x2 2x+4, 2<x23 -4x, 23<x0 4x , 0 <x2 2x+4 , x>2
The graph of y = f(x) is as follows

From graph f(x) has local minimum at -2, 0 and local maximum at 23


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