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If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II) 25 Mar 2014, 14:06
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VeritasPrepKarishma
If f(x) = |4x - 1| + |x-3| + |x + 1|, what is the minimum value of f(x)?

(A) 3
(B) 4
(C) 5
(D) 21/4
(E) 7

(Still high on mods! Next week, will make questions on some other topic.)
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Here is the easiest solution
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If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II) 14 Feb 2016, 06:41
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Is it safe to say that for this type of questions one just needs to test checkpoints (-1; 1/4; 3) to find out the min value? Just because by utilizing one of those points we will get rid of one "trip" and hence bascially the total length of trips would be shorter (when one person stays at home as Karishma exemplified).
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If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II) Updated on: 05 Dec 2023, 02:53
Originally posted by KarishmaB on 16 Feb 2016, 23:20.
Last edited by KarishmaB on 05 Dec 2023, 02:53, edited 2 times in total.
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shasadou
Is it safe to say that for this type of questions one just needs to test checkpoints (-1; 1/4; 3) to find out the min value? Just because by utilizing one of those points we will get rid of one "trip" and hence bascially the total length of trips would be shorter (when one person stays at home as Karishma exemplified).
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Yes, the game changer will be at the transition point.If one or more people stay at home, the length of the trip shortens.

This video discusses the basics of absolute values: https://youtu.be/oqVfKQBcnrs
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If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II) 23 Jul 2019, 19:01
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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
answer:4

VeritasKarishma Any wrong here?
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If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II) 23 Jul 2019, 23:25
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ShivamoggaGaganhs
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
answer:4

VeritasKarishma Any wrong here?
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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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If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II) 23 Apr 2020, 05:26
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VeritasKarishma
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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?'
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Hi VeritasKarishma
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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If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II) 23 Apr 2020, 22:48
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ultimateashish
VeritasKarishma
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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?'

Hi VeritasKarishma
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?
Show more


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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If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II) 07 Jan 2026, 08:06
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for any function like |x + a| + |x - b| +..
there will be a minima and it will be one of the point at -a, b etc
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