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# If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II)

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Re: If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II) [#permalink]
The min value of the mod is 0

make individual mods equal to zero.

We can get
x=1/4
x=3
x=-1

Then, substitute these values one by one to see which value of x gives you the least
You will get it for x=1/4

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Re: If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II) [#permalink]
ShivamoggaGaganhs wrote:
The min value of the mod is 0

make individual mods equal to zero.

We can get
x=1/4
x=3
x=-1

Then, substitute these values one by one to see which value of x gives you the least
You will get it for x=1/4

With 3 points, and co-efficient of x as 1 for each term, the minimum will be at the middle point on the number line.
With 4 points, the minimum would be in a range.
It's best to understand the process and why it is followed to make adjustments as needed.
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Re: If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II) [#permalink]
eaakbari wrote:
Karishma,

What about a function with evenly spaced numbers?

for instance

f(x) = | 4x + 1| + | 2x + 1| + | 4x + 3| + | x |

What will be the min. value of x for this?

The function will take a minimum value for a range of values of x (between the second and the third values). Think 'Why?'

Regarding the question given above, using your approach, shouldn't the minimum value be at x = -1/2 ?
Taking critical points as -1/4, -1/2, -3/4 and 0. So, there are four people each at -1/4 and -3/4, and 2 people at -1/2 and one person at 0.
Representing on number line as follows.

(4)........(2)...........(4)........(1)
-3/4......-1/2........-1/4..........0

So, 4 guys at -3/4 and 4 guys at -1/4 come to -1/2. Now there are 10 people at -1/2 and 1 person at 0. So minimum f(x) should be at x = -1/2.
Had they been not equidistant then the value of x could have been in a range for f(x) to be minimum?
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Re: If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II) [#permalink]
ultimateashish wrote:
eaakbari wrote:
Karishma,

What about a function with evenly spaced numbers?

for instance

f(x) = | 4x + 1| + | 2x + 1| + | 4x + 3| + | x |

What will be the min. value of x for this?

The function will take a minimum value for a range of values of x (between the second and the third values). Think 'Why?'

Regarding the question given above, using your approach, shouldn't the minimum value be at x = -1/2 ?
Taking critical points as -1/4, -1/2, -3/4 and 0. So, there are four people each at -1/4 and -3/4, and 2 people at -1/2 and one person at 0.
Representing on number line as follows.

(4)........(2)...........(4)........(1)
-3/4......-1/2........-1/4..........0

So, 4 guys at -3/4 and 4 guys at -1/4 come to -1/2. Now there are 10 people at -1/2 and 1 person at 0. So minimum f(x) should be at x = -1/2.
Had they been not equidistant then the value of x could have been in a range for f(x) to be minimum?

Yes, the function will be minimum at x = -1/2 here. The co-efficients are not the same. With a 2 at extreme right, it will become a range again.
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Re: If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II) [#permalink]
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Re: If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II) [#permalink]
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