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sujoykrdatta
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Minimum value of the function f(x) = |x – a| + |x – b| + |x – c| 14 May 2021, 12:46
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95% (hard)

Question Stats:

34% (02:03) correct 66% (01:57) wrong based on 184 sessions

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What is the minimum value of the function f(x) = |x – a| + |x – b| + |x – c|, where a, b, c are distinct numbers?

(1) 0< a < b < c
(2) c - a = 10
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C
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chetan2u
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Minimum value of the function f(x) = |x – a| + |x – b| + |x – c| 14 May 2021, 20:30
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sujoykrdatta
What is the minimum value of the function f(x) = |x – a| + |x – b| + |x – c|, where a, b, c are distinct numbers?

(1) 0< a < b < c
(2) c - a = 10
Show more


The function is |x – a| + |x – b| + |x – c|.

Whenever x is between the extreme two points, the value of function will be less than when x is greater or lesser than all three variables, a, b and c.
And it will be minimum, when x is equal to middle value as it will lead to one MOD being 0.

To explain it further, I have added a sketch.
There are 4 cases:
1) Case 1: When x<a<b<c and is shown by brown line.
The value will be the sum of brown arrows, and will be (c-a)+(b-a)+3(a-x)
2) Case 2: When a<x<b<c and is shown by blue line.
The value will be the sum of blue arrows, and will be (c-a)+|b-x|
3) Case 3: When a<b=x<c and is shown by red line.
The value will be the sum of red arrows, and will be (c-a)
4) Case 4: When a<b<c<x and is shown by green line.
The value will be the sum of green arrows, and will be (c-a)+(c-b)+3(x-a)
Clearly case 3 gives the minimum value.

(1) 0< a < b < c
Now we know that the value will be the minimum when x=b, as b is the middle value.
So, |x – a| + |x – b| + |x – c| = |b – a| + |b – b| + |b – c| = |b - a| + |b - c|
Now
b>a, so b-a>0 and
c>b, so b-c<0
Hence, \(|b-a|+|b-c|=b-a+c-b=c-a\)
But we do not know value of a and c.
Insufficient

(2) c - a = 10
Nothing about the values
Insufficient

Combined
From statement I, the minimum value is c-a, and statement II gives you c-a=10.
Hence, the minimum value is 10.
Sufficient


C
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Minimum value of the function f(x) = |x – a| + |x – b| + |x – c| 14 May 2021, 23:05
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chetan2u
sujoykrdatta
What is the minimum value of the function f(x) = |x – a| + |x – b| + |x – c|, where a, b, c are distinct numbers?

(1) 0< a < b < c
(2) c - a = 10


The function is |x – a| + |x – b| + |x – c|.

Whenever x is between the extreme two points, the value of function will be less than when x is greater or lesser than all three variables, a, b and c.
And it will be minimum, when x is equal to middle value as it will lead to one MOD being 0.

(1) 0< a < b < c
Now we know that the value will be the minimum when x=b, as b is the middle value.
So, |x – a| + |x – b| + |x – c| = |b – a| + |b – b| + |b – c| = |b - a| + |b - c|
Now
b>a, so b-a>0 and
c>b, so b-c<0
Hence, \(|b-a|+|b-c|=b-a+c-b=c-a\)
But we do not know value of a and c.
Insufficient

(2) c - a = 10
Nothing about the values
Insufficient

Combined
From statement I, the minimum value is c-a, and statement II gives you c-a=10.
Hence, the minimum value is 10.
Sufficient


C
Show more


Hello chetan2u sir,

My question is w.r.t the statement: Whenever x is between the extreme two points, the value of function will be less than when x is greater or lesser than all three variables, a, b and c. And it will be minimum, when x is equal to middle value as it will lead to one MOD being 0.

I plugged in some values in the function and found that this statement is correct. Are there more such properties that we need to learn/understand as far is MOD is concerned?
Also, I could not wrap my head around the below part either:

Now we know that the value will be the minimum when x=b, as b is the middle value.
So, |x – a| + |x – b| + |x – c| = |b – a| + |b – b| + |b – c| = |b - a| + |b - c|
Now
b>a, so b-a>0 and
c>b, so b-c<0
Hence, \(|b-a|+|b-c|=b-a+c-b=c-a\)

Is it possible for you to explain it more elaborately?

Sorry to bother you.
Thanks.
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Minimum value of the function f(x) = |x – a| + |x – b| + |x – c| 15 May 2021, 02:07
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beeblebrox
chetan2u
sujoykrdatta
What is the minimum value of the function f(x) = |x – a| + |x – b| + |x – c|, where a, b, c are distinct numbers?

(1) 0< a < b < c
(2) c - a = 10


The function is |x – a| + |x – b| + |x – c|.

Whenever x is between the extreme two points, the value of function will be less than when x is greater or lesser than all three variables, a, b and c.
And it will be minimum, when x is equal to middle value as it will lead to one MOD being 0.

(1) 0< a < b < c
Now we know that the value will be the minimum when x=b, as b is the middle value.
So, |x – a| + |x – b| + |x – c| = |b – a| + |b – b| + |b – c| = |b - a| + |b - c|
Now
b>a, so b-a>0 and
c>b, so b-c<0
Hence, \(|b-a|+|b-c|=b-a+c-b=c-a\)
But we do not know value of a and c.
Insufficient

(2) c - a = 10
Nothing about the values
Insufficient

Combined
From statement I, the minimum value is c-a, and statement II gives you c-a=10.
Hence, the minimum value is 10.
Sufficient


C


Hello chetan2u sir,

My question is w.r.t the statement: Whenever x is between the extreme two points, the value of function will be less than when x is greater or lesser than all three variables, a, b and c. And it will be minimum, when x is equal to middle value as it will lead to one MOD being 0.

I plugged in some values in the function and found that this statement is correct. Are there more such properties that we need to learn/understand as far is MOD is concerned?
Also, I could not wrap my head around the below part either:

Now we know that the value will be the minimum when x=b, as b is the middle value.
So, |x – a| + |x – b| + |x – c| = |b – a| + |b – b| + |b – c| = |b - a| + |b - c|
Now
b>a, so b-a>0 and
c>b, so b-c<0
Hence, \(|b-a|+|b-c|=b-a+c-b=c-a\)

Is it possible for you to explain it more elaborately?

Sorry to bother you.
Thanks.
Show more


Hi. Think it of like this:
On the number line, mark 3 points: a, b, c

----|----|----|----
----a----b----c----

Take x on the extreme right
Use the logic: |x-c| means the distance between x and c

Thus: |x – a| + |x – b| + |x – c| means
Sum of the distances of x from a, x from b and x from c.
Observe that there are multiple overlaps.

As you bring x more to the left, approaching b, the overlaps keep decreasing (draw such diagrams to understand)
When x lands on b, there is no overlap and the sum of the distances simply becomes the distance between a and c (the extreme points).


Note: if there were 4 points:
----|----|----|----|----
----a----b----c----d----

Then, the minimum sum will occur when x is between b and c (inclusive)

Try drawing and you will understand.
If not, let me know, I'll upload a sketch.

Posted from my mobile device
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Minimum value of the function f(x) = |x – a| + |x – b| + |x – c| 15 May 2021, 07:22
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beeblebrox
chetan2u
sujoykrdatta
What is the minimum value of the function f(x) = |x – a| + |x – b| + |x – c|, where a, b, c are distinct numbers?

(1) 0< a < b < c
(2) c - a = 10


The function is |x – a| + |x – b| + |x – c|.

Whenever x is between the extreme two points, the value of function will be less than when x is greater or lesser than all three variables, a, b and c.
And it will be minimum, when x is equal to middle value as it will lead to one MOD being 0.

(1) 0< a < b < c
Now we know that the value will be the minimum when x=b, as b is the middle value.
So, |x – a| + |x – b| + |x – c| = |b – a| + |b – b| + |b – c| = |b - a| + |b - c|
Now
b>a, so b-a>0 and
c>b, so b-c<0
Hence, \(|b-a|+|b-c|=b-a+c-b=c-a\)
But we do not know value of a and c.
Insufficient

(2) c - a = 10
Nothing about the values
Insufficient

Combined
From statement I, the minimum value is c-a, and statement II gives you c-a=10.
Hence, the minimum value is 10.
Sufficient


C


Hello chetan2u sir,

My question is w.r.t the statement: Whenever x is between the extreme two points, the value of function will be less than when x is greater or lesser than all three variables, a, b and c. And it will be minimum, when x is equal to middle value as it will lead to one MOD being 0.

I plugged in some values in the function and found that this statement is correct. Are there more such properties that we need to learn/understand as far is MOD is concerned?
Also, I could not wrap my head around the below part either:

Now we know that the value will be the minimum when x=b, as b is the middle value.
So, |x – a| + |x – b| + |x – c| = |b – a| + |b – b| + |b – c| = |b - a| + |b - c|
Now
b>a, so b-a>0 and
c>b, so b-c<0
Hence, \(|b-a|+|b-c|=b-a+c-b=c-a\)

Is it possible for you to explain it more elaborately?

Sorry to bother you.
Thanks.
Show more


beeblebrox

Added few more details in the original post. Please see if it helps you.
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Minimum value of the function f(x) = |x – a| + |x – b| + |x – c| 07 Jan 2025, 12:16
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