How Can You Minimize the Expression Involving Absolute Values?

[anemone]

Here is this week's POTW:

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Minimize $|x+2|+2|x-5|+|2x-7|+|0.5x-5.5|$, given $x$ is a real number.

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

[anemone]

Congratulations to the following members for their correct solution::)

1. kaliprasad
2. greg1313

Solution from kaliprasad:

Spoiler

Let the function be f(x)

we have

$f(x)=|x+2|+2|x−5|+|2x−7|+|.5x−5.5|$

we need to consider at 4 points

$x=−2,x=3.5,x=5.x=11$

now we get the f(x) as in 3 regions as

$f(x)=−3.5x+24.5$ for $−2≤x≤3.5$

$f(x)=.5x+10.5$ for $3.5≤x≤5$

$f(x)=4.5x−9.5$ for $5≤x≤11$

Clearly $f(x)$ decreases from $x = - 2$ to $3.5$ and increases from $3.5$ to $11$.

Before $-2$ $f(x)$ decreases in range from - infinity to $- 2$ and it increases in range from $11$ to infinity and hence the lowest point occurs at $x=3.5$ the value being $f(3.5)=12.25$.