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14 May 2021, 12:46
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Difficulty:
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Question Stats:
34% (02:04) correct
66%
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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
(1) 0< a < b < c
(2) c - a = 10
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
(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.
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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General Discussion
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
(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.
