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If a > b > c > 0, then what is the value of the minimum value of the function f(x) = |x - a| + |x - b| + |x - c| ?

A. 2a - b - c
B. a + b - 2c
C. a + b + c
D. a - c
E. a + c


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Bunuel
If a > b > c > 0, then what is the value of the minimum value of the function f(x) = |x - a| + |x - b| + |x - c| ?

A. 2a - b - c
B. a + b - 2c
C. a + b + c
D. a - c
E. a + c

Show more

Solution:

If x = a, then f(a) = 0 + (a - b) + (a - c) = (a - b) + (a - c).

If x = b, then f(b) = -(b - a) + 0 + (b - c) = a - c.

If x = c, then f(c) = -(c - a) + [-(c - b)] + 0 = (a - c) + (b - c).

We see that choice A is f(a), choice B is f(c) and choice D is f(b). Furthermore, we see that both f(a) and f(c) are greater than f(b) because they both have a positive term besides a - c. Since both choices C and E are also greater than a - c, we see that f(b) = a - c must be the minimum value of f(x).

Answer: D
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If a > b > c > 0, then what is the value of the minimum value of the 15 Jun 2020, 00:15
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Bunuel
If a > b > c > 0, then what is the value of the minimum value of the function f(x) = |x - a| + |x - b| + |x - c| ?

A. 2a - b - c
B. a + b - 2c
C. a + b + c
D. a - c
E. a + c

Are You Up For the Challenge: 700 Level Questions
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Always remember that when you have x between the two extreme points, the value of the modulus will be the difference between these two extreme points.

So \(|x-a|+|x-c|\) will have minimum value as \(a-c\) whenever \(c<x<a\).

But we are looking for minimum value of \(|x - a| + |x - b| + |x - c| \) where \(a>b>c>0\).
Now minimum value of \(|x - a| + |x - b| + |x - c| \) will be a-c+|x-b|, (as \(|x-a|+|x-c|\) will have minimum value as \(a-c\) whenever \(c<x<a\)) and
|x-b|=0 when x=b and also then x or b lies between a and c.
So \(a-c+0=a-c\)

D­
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If a > b > c > 0, then what is the value of the minimum value of the 27 May 2020, 05:19
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Given: f(x) = |x - a| + |x - b| + |x - c| & a > b > c > 0

f(x) will be minimum when x=b

so f(x)min = |b-a| + |b-b| + |b-c| = 2b-a-c but that is not a option.

I also tried assuming a=3, b=2, c= 1 and gets min value when x= b or x = 2.

Can somebody guide.
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ankurjuit
if you observe here """ so f(x)min = |b-a| + |b-b| + |b-c| = 2b-a-c but that is not a option. """"
|b-a| = a-b as a>b
so your expression will be a-b+b-c which is a-c option d.
Does it clear your doubt?
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If a > b > c > 0, then what is the value of the minimum value of the 27 May 2020, 05:47
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If a > b > c > 0, then what is the value of the minimum value of the function f(x) = |x - a| + |x - b| + |x - c| ?

A. 2a - b - c
B. a + b - 2c
C. a + b + c
D. a - c
E. a + c

Show more

Solution


    • a > b > c > 0
      o It means a, b, and c are distinct positive numbers.
    • Now, f(x) = |x – a| + |x – b| + |x – c|
      o Since, three absolute values are added, so f(x) ≥ 0
         f(x) = 0 ⟹ |x- a| = |x – b| = |x – c|= 0 ⟹ x = a = b= c, which is not possible.
         So f(x) cannot be 0, but we will try to make at least one term 0.
    • Since, b will come in between a and c on the number line. (a – b) or (b-c ) < (a – c)
      o Therefore, f(x) will be minimum at x = b
      o f(x) = | b-a| + |b – b| + |b-c| = - (b- a) + 0 + b – c = a- c
Thus, the correct answer is Option D.
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If a > b > c > 0, then what is the value of the minimum value of the 27 May 2020, 10:47
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See the attachment -> D
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If a > b > c > 0, then what is the value of the minimum value of the 28 May 2020, 07:43
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Bunuel
If a > b > c > 0, then what is the value of the minimum value of the function f(x) = |x - a| + |x - b| + |x - c| ?

A. 2a - b - c
B. a + b - 2c
C. a + b + c
D. a - c
E. a + c

Are You Up For the Challenge: 700 Level Questions
Show more
On a number line from left to right critical points are c....b.....a
..(-).......c.........(+)........ b.........(-)...........a.....(+)........
Since F(x) is positive, for values of F(x) we will look into (+) interval
Greater than "a" or Between c and b
For X>=a
F(X)=0+x-b+x-c
=2x-b-c=2a-b-c=(a-b)+(a-c)=(a-c)+ positive value
For c<x<=b
F(x)=-x+a+0+x-c
F(x)=a-c
D :)­
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If a > b > c > 0, then what is the value of the minimum value of the 10 Jun 2020, 10:51
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Bunuel
If a > b > c > 0, then what is the value of the minimum value of the function f(x) = |x - a| + |x - b| + |x - c| ?

A. 2a - b - c
B. a + b - 2c
C. a + b + c
D. a - c
E. a + c

Are You Up For the Challenge: 700 Level Questions
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a > b > c > 0, minimum value of f(x) = |x - a| + |x - b| + |x - c| ?
Visualize this on the number line
|x - a| + |x - b| + |x - c| this is sum of distances of a number x from a , b and c, if x is greater than a or less than c the sum of distances will keep increasing.
and sum of distances from b will be the least, as it lies between the other two numbers,

we can check by substitution:
distance from a would be : 0+ |a-b| + | a-c| = a-b+a-c=2a-b-c
distance from b : |b-a|+0+|b-c| = a-b+b-c = a-c
distnace from c: |c-a|+|c-b|+0 = a-c+b-c = a+b - 2c
and from these we know a-b >0 (since a>b) so 2a-b-c = a-c+ (a-b) > a-c
similarly with b-c>0 (since b>c) so a+b-2c = a-c + (b-c) > a-c
thus we see the minimum value is a-c (when x=b )
D­
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If a > b > c > 0, then what is the value of the minimum value of the 13 Jun 2020, 04:46
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Hi,

Can anyone explain why |b-a| becomes -b+a?

I also ended up with 2b -a - c

Thanks a lot :)
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If a > b > c > 0, then what is the value of the minimum value of the 14 Jun 2020, 05:34
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ScottTargetTestPrep
Bunuel
If a > b > c > 0, then what is the value of the minimum value of the function f(x) = |x - a| + |x - b| + |x - c| ?

A. 2a - b - c
B. a + b - 2c
C. a + b + c
D. a - c
E. a + c


Solution:

If x = a, then f(a) = 0 + (a - b) + (a - c) = (a - b) + (a - c).

If x = b, then f(b) = -(b - a) + 0 + (b - c) = a - c.

If x = c, then f(c) = -(c - a) + [-(c - b)] + 0 = (a - c) + (b - c).

We see that choice A is f(a), choice B is f(c) and choice D is f(b). Furthermore, we see that both f(a) and f(c) are greater than f(b) because they both have a positive term besides a - c. Since both choices C and E are also greater than a - c, we see that f(b) = a - c must be the minimum value of f(x).

Answer: D

Show more


hey ScottTargetTestPrep :)

Would you please take your time to RE-explain your solution :grin:

i don`t get why you put minus in from of parentheses .... so here is my question: why ? :)
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If a > b > c > 0, then what is the value of the minimum value of the 14 Jun 2020, 05:37
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Bunuel
If a > b > c > 0, then what is the value of the minimum value of the function f(x) = |x - a| + |x - b| + |x - c| ?

A. 2a - b - c
B. a + b - 2c
C. a + b + c
D. a - c
E. a + c

Are You Up For the Challenge: 700 Level Questions
On a number line from left to right critical points are c....b.....a
..(-).......c.........(+)........ b.........(-)...........a.....(+)........
Since F(x) is positive, for values of F(x) we will look into (+) interval
Greater than "a" or Between c and b
For X>=a
F(X)=0+x-b+x-c
=2x-b-c=2a-b-c=(a-b)+(a-c)=(a-c)+ positive value
For c<x<=b
F(x)=-x+a+0+x-c
F(x)=a-c
D :)
Show more
hey satya2029 can you please explain what method are you using ? :? by ..(-).......c.........(+)........ b.........(-)...........a.....(+)........ is it wavy curve ? or what i am kinda confused
thanks :)­
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sambitspm
See the attachment -> D
Show more

hey sambitspm
man, looking at your fantastic painting how did you arrive at the answer ? :) is wayvy curve or pyramid method ...:grin: i kinda cant wrap my mind around your solution, would appreciate your answer :)

p.s. Just imagine in 100 years this painting will cost 5 million EUR :lol: :lol: :lol:
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If a > b > c > 0, then what is the value of the minimum value of the Updated on: 14 Jun 2020, 12:31
Originally posted by satya2029 on 14 Jun 2020, 06:43.
Last edited by satya2029 on 14 Jun 2020, 12:31, edited 1 time in total.
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Bunuel
If a > b > c > 0, then what is the value of the minimum value of the function f(x) = |x - a| + |x - b| + |x - c| ?

A. 2a - b - c
B. a + b - 2c
C. a + b + c
D. a - c
E. a + c

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Are You Up For the Challenge: 700 Level Questions

On a number line from left to right critical points are c....b.....a
..(-).......c.........(+)........ b.........(-)...........a.....(+)........
Since F(x) is positive, for values of F(x) we will look into (+) interval
Greater than "a" or Between c and b
For X>=a
F(X)=0+x-b+x-c
=2x-b-c=2a-b-c=(a-b)+(a-c)=(a-c)+ positive value
For c<x<=b
F(x)=-x+a+0+x-c
F(x)=a-c
D:)

hey satya2029 can you please explain what method are you using ? :? by ..(-).......c.........(+)........ b.........(-)...........a.....(+)........ is it wavy curve ? or what i am kinda confused
thanks :)
Show more
Sure.
Consider each modulus element as a separate function
Since at X=a , lX-al=0, a is a critical points, so are b and c
plot them on a number line.
From extreme right here "a" when x>=a, assign a positive sign and then change the sign alternatively
as here a>b>c
......(-)........c.....(+).......b.......(-)...........a(+)
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Bunuel
If a > b > c > 0, then what is the value of the minimum value of the function f(x) = |x - a| + |x - b| + |x - c| ?

A. 2a - b - c
B. a + b - 2c
C. a + b + c
D. a - c
E. a + c


Are You Up For the Challenge: 700 Level Questions
Show more
dave13

We know that f(x) = |x - a| + |x - b| + |x - c|

i.e .f(x) = Distance of x from a + Distance of x from b + Distance of x from c

for minimizing this expression, x must be equal to b because b is between a and c and x will be closest to both a, b and c



For x = b

f(x) = |x - a| + |x - b| + |x - c| = -x+a + 0 + x-c = a-c

Answer: Option D­
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If a > b > c > 0, then what is the value of the minimum value of the Updated on: 18 Jun 2024, 09:53
Originally posted by Kinshook on 14 Jun 2020, 09:36.
Last edited by Bunuel on 18 Jun 2024, 09:53, edited 2 times in total.
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Bunuel
If a > b > c > 0, then what is the value of the minimum value of the function f(x) = |x - a| + |x - b| + |x - c| ?

A. 2a - b - c
B. a + b - 2c
C. a + b + c
D. a - c
E. a + c


Are You Up For the Challenge: 700 Level Questions
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Asked: If a > b > c > 0, then what is the value of the minimum value of the function f(x) = |x - a| + |x - b| + |x - c| ?

f(x) = |x - a| + |x - b| + |x - c|

Case 1: a>b>c>={x,0}
f(x) = |x - a| + |x - b| + |x - c| = (a-x) + (b-x) + (c-x) = (a+b+c-3x)
f(0) = a + b + c
f(c) = a + b - 2c

Case 2: a> b>= x > c > 0
f(x) = |x - a| + |x - b| + |x - c| = (a-x) + (b-x) + (x-c) = (a+b-x-c)
f(b) = a - c

Case 3: a>=x>b>c>0
f(x) = |x - a| + |x - b| + |x - c| = (a-x) + (x-b) + (x-c) = (a-b+x-c)
f(a) = 2a - b - c

a + b - 2c > a - c ; Since a + b - 2c = a + (b-c) - c > a -c ; Since b-c>0
2a - b - c = a + (a-b) - c > a - c ; Since (a-b)>0

a-c is the minimum value of the function f(x) = |x - a| + |x - b| + |x - c|

IMO D­
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Bunuel
If a > b > c > 0, then what is the value of the minimum value of the function f(x) = |x - a| + |x - b| + |x - c| ?

A. 2a - b - c
B. a + b - 2c
C. a + b + c
D. a - c
E. a + c


Are You Up For the Challenge: 700 Level Questions
dave13

We know that f(x) = |x - a| + |x - b| + |x - c|

i.e .f(x) = Distance of x from a + Distance of x from b + Distance of x from c

for minimizing this expression, x must be equal to b because b is between a and c and x will be closest to both a, b and c



For x = b

f(x) = |x - a| + |x - b| + |x - c| = -x+a + 0 + x-c = a-c

Answer: Option D
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many thanks GMATinsight :) appreciate your taking time to answer my question :)
just one more question why did you put minus sign in front of x ? :? (see highlighted part) are checking one case only ? i mean to break down by cases openening modulus and etc :grin:­
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Bunuel
If a > b > c > 0, then what is the value of the minimum value of the function f(x) = |x - a| + |x - b| + |x - c| ?

A. 2a - b - c
B. a + b - 2c
C. a + b + c
D. a - c
E. a + c


Are You Up For the Challenge: 700 Level Questions
dave13

We know that f(x) = |x - a| + |x - b| + |x - c|

i.e .f(x) = Distance of x from a + Distance of x from b + Distance of x from c

for minimizing this expression, x must be equal to b because b is between a and c and x will be closest to both a, b and c



For x = b

f(x) = |x - a| + |x - b| + |x - c| = -x+a + 0 + x-c = a-c

Answer: Option D
many thanks GMATinsight :) appreciate your taking time to answer my question :)
just one more question why did you put minus sign in front of x ? :? (see highlighted part) are checking one case only ? i mean to break down by cases openening modulus and etc :grin:
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dave13

Because a > b > c and x = b

i.e. a > x

then lx-al = a-x

e.g. @a=5, x=2
lx-al = l2-5l = l-3l = 3 which is same as a-x or written as -x+a :)­
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If a > b > c > 0, then what is the value of the minimum value of the 29 Jun 2020, 08:56
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dave13
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Bunuel
If a > b > c > 0, then what is the value of the minimum value of the function f(x) = |x - a| + |x - b| + |x - c| ?

A. 2a - b - c
B. a + b - 2c
C. a + b + c
D. a - c
E. a + c


Solution:

If x = a, then f(a) = 0 + (a - b) + (a - c) = (a - b) + (a - c).

If x = b, then f(b) = -(b - a) + 0 + (b - c) = a - c.

If x = c, then f(c) = -(c - a) + [-(c - b)] + 0 = (a - c) + (b - c).

We see that choice A is f(a), choice B is f(c) and choice D is f(b). Furthermore, we see that both f(a) and f(c) are greater than f(b) because they both have a positive term besides a - c. Since both choices C and E are also greater than a - c, we see that f(b) = a - c must be the minimum value of f(x).

Answer: D



hey ScottTargetTestPrep :)

Would you please take your time to RE-explain your solution :grin:

i don`t get why you put minus in from of parentheses .... so here is my question: why ? :)
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Response:

If x = b, then f(b) = |b - a| + |b - b| + |b - c|. Obviously, |b - b| = 0. For |b - a|, we are told that a > b > c > 0, thus b - a < 0. For an expression less than zero, absolute value is found by multiplying the expression by -1, hence |b - a| = -(b - a). That’s why I put minus in front of the parentheses for the first highlighted expression. Also, b - c > 0 and thus, |b - c| = b - c. That’s why I did not put any minus signs in front of the parentheses in front of the corresponding expression.

For the second and third highlights, using the same arguments, c - a < 0 and thus, |c - a| = -(c - a). Similarly, c - b < 0 and thus, |c - b| = -(c - b).
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If a > b > c > 0, then what is the value of the minimum value of the 30 Jun 2022, 04:44
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NitishJain
Given: f(x) = |x - a| + |x - b| + |x - c| & a > b > c > 0

f(x) will be minimum when x=b

so f(x)min = |b-a| + |b-b| + |b-c| = 2b-a-c but that is not a option.

I also tried assuming a=3, b=2, c= 1 and gets min value when x= b or x = 2.

Can somebody guide.
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f(x) will be minimum when x=b is the perfect intuition to have. When you get here, it becomes easier in my opinion!

If x=b, then the first term, which is the distance btw a and x, becomes: a-b
If x=b, then the second term, which is the distance btw b and x, becomes: 0
If x=b, then the third term, which is the distance btw c and x, becomes: b-c

(a-b)+(b-c) = a-c
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