A problem that cannot make up its mind

Algebra Level 3

$Y = |x| - |x + 1| + |x + 2| - |x + 3| + \cdots +|x + 2016|$

Find the minimum value of $Y$ .

Notation : $| \cdot |$ denotes the absolute value function .

The answer is 1008.

3 solutions

Shekhar Prasad
Aug 12, 2016

Consider the above expression as strings connecting all integer points -2016,-2015...... till 0. We are here asked to minimize the length of string. If we start tying the strings from point -1008 in both directions we achieve our condition of minimum length.

We get the above expression stated as under:-

Y = 1008-1007+1006-1005+....+0 -1+2-3+4+....1008 = 1008

Daria Zafote
Apr 14, 2018

By looking at the expression we can see that it can’t be negative - to make this more clear we can group the elements like this:

Y = [(|x|+|x+2016|) - (|x+1|+|x+2015|)] + [(|x+2|+|x+2014|) - (|x+3|+|x+2013|)] + [(|x+4|+|x+2012|) - (|x+5|+|x+2011|)] ... + |x+1008|

At the end we get to +|x+1008| which stands alone. We can minimize this expression by choosing x that will make the pairs in the square brackets cancel. A perfect number for this is 0 which leaves us with +|0+1008|=1008 so this is the minimum value.

Why is it stand alone you ask? (Not you Daria)

There're odd number of terms here, so we can pair every modulus except $|x+1008|$

Akshay Krishna - 2 years, 6 months ago

Anu Rag
Sep 12, 2017

Take a small set of elements y=|x|-|x+1|+|x+2| now the min value of y is 1; Either by x=(0+2)/2=1 or by x=0 and nothing else, so when we substitute any two values as x=0 or x=1008,we get the minimum as 1008

A problem that cannot make up its mind

The answer is 1008.

3 solutions

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