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Re: If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II) [#permalink]

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09 Jul 2013, 19:58

Here is my idea: f(x) = f1(x) + f2(x) + f3(x) with all the functions are first-order ones. Suppose that: f1(x1)=0 --> x1=1/4 f2(x2)=0 --> x2=3 f3(x3)=0 --> x3=-1 Always does f(x) reach its minimum value with x(1) or (x2) or (x3) in this problem, having absolute value | | Therefore, we can calculate the value of f(1/4), f(3), f(-1) f(1/4)=4 f(3) and f(-1) are both bigger than 4 surely without further calculation

That minimum of a function will always be at either its critical points or zero.

Await your valued response.

The minimum could also be in an entire range. Take this question for example.

f(x) = |3x + 1| + |2x-3| + |x - 7|

For what value(s) of x will f(x) have the minimum value?

Thank you for your intuitive explanation and clearing the concept for us. So using your analogy of people at every point, shouldn't -1/3 have the minimum value for x (or distance so to speak)?

To be precise, you said 1/4 point has the minimum distance for x or gives the minimum value because of the denominator 4 (my assumption - as you did not state it specifically). This is why we chose 1/4 over 3/2!
_________________

Our deepest fear is not that we are inadequate. Our deepest fear is that we are powerful beyond measure. It is our light not our darkness that most frightens us.

Your playing small does not serve the world. There's nothing enlightened about shrinking so that other people won't feel insecure around you.

It's not just in some of us; it's in everyone. And as we let our own light shine, we unconsciously give other people permission to do the same.

As we are liberated from our own fear, our presence automatically liberates others.

That minimum of a function will always be at either its critical points or zero.

Await your valued response.

The minimum could also be in an entire range. Take this question for example.

f(x) = |3x + 1| + |2x-3| + |x - 7|

For what value(s) of x will f(x) have the minimum value?

Thank you for your intuitive explanation and clearing the concept for us. So using your analogy of people at every point, shouldn't -1/3 have the minimum value for x (or distance so to speak)?

To be precise, you said 1/4 point has the minimum distance for x or gives the minimum value because of the denominator 4 (my assumption - as you did not state it specifically). This is why we chose 1/4 over 3/2!

f(x) = |3x + 1| + |2x-3| + |x - 7|

f(x) = 3|x + 1/3| + 2|x-3/2| + |x - 7|

-1/3 ------------------ 3/2-----------------------------------7 (3) ...........................(2)...........................................(1) Say, there are 3 people at -1/3, 2 people at 3/2 and 1 person at 7. They need to meet while covering the least distance. Where should they meet?

Obviously, the person at 7 should travel to 3/2. The distance covered will be 7 - 3/2 = 11/2 Now there are 3 people at -1/3 and 3 people at 3/2. They can meet anywhere between -1/3 and 3/2. The distance covered will be the same in each case.

The point is not whether it is 1/4, the point is the constant outside i.e. how many people need to travel from that point.
_________________

For what value(s) of x will f(x) have the minimum value?

Thank you for your intuitive explanation and clearing the concept for us. So using your analogy of people at every point, shouldn't -1/3 have the minimum value for x (or distance so to speak)?

To be precise, you said 1/4 point has the minimum distance for x or gives the minimum value because of the denominator 4 (my assumption - as you did not state it specifically). This is why we chose 1/4 over 3/2!

f(x) = |3x + 1| + |2x-3| + |x - 7|

f(x) = 3|x + 1/3| + 2|x-3/2| + |x - 7|

-1/3 ------------------ 3/2-----------------------------------7 (3) ...........................(2)...........................................(1) Say, there are 3 people at -1/3, 2 people at 3/2 and 1 person at 7. They need to meet while covering the least distance. Where should they meet?

Obviously, the person at 7 should travel to 3/2. The distance covered will be 7 - 3/2 = 11/2 Now there are 3 people at -1/3 and 3 people at 3/2. They can meet anywhere between -1/3 and 3/2. The distance covered will be the same in each case.

The point is not whether it is 1/4, the point is the constant outside i.e. how many people need to travel from that point.

Thank you for you prompt reply. As we use the people analogy, you mentioned that 2 people are at point 3/2 and 3 are at -1/3. Wouldn't the distance covered be more if 3 people had to travel instead of 2: because the constant outside the mod is 3, true?

Alternatively, if I use the traditional approach i.e. calculate the distance among these three points, I get to the conclusion that point 3/2 is where they cover the least distance.

So what am I missing here? Please assist me in getting the concept right using your holistic approach to Mods. Thanks.
_________________

Our deepest fear is not that we are inadequate. Our deepest fear is that we are powerful beyond measure. It is our light not our darkness that most frightens us.

Your playing small does not serve the world. There's nothing enlightened about shrinking so that other people won't feel insecure around you.

It's not just in some of us; it's in everyone. And as we let our own light shine, we unconsciously give other people permission to do the same.

As we are liberated from our own fear, our presence automatically liberates others.

Re: If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II) [#permalink]

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10 Sep 2013, 21:12

In the interval (-1,1/4) f(x)=-(4x-1)-(x-3)+(x+1)=-4x+1 f(0)=1 However, when I plug in 0 in the original f(x), I get it it to equal 5. f(0)=5. What am I doing wrong?

In the interval (-1,1/4) f(x)=-(4x-1)-(x-3)+(x+1)=-4x+1 f(0)=1 However, when I plug in 0 in the original f(x), I get it it to equal 5. f(0)=5. What am I doing wrong?

Re: If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II) [#permalink]

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14 Oct 2013, 05:18

May I ask a couple of silly questions : why do we find critical values when solving for such questions (with multiple mod expressions, either equalities or inequalities)?

And why do we find critical values equating the expressions inside the mod to zero? Is this because we know that the expression must be equal to or greater than zero?

Can't quite explain it to myself. Please help me out to straighten these out.
_________________

There are times when I do not mind kudos...I do enjoy giving some for help

I am wondering how can this method be used in questions where there are negative between terms : Ex: minimum value of : |x+6|-|x-1| ?

You can do it with a negative sign too. You want to find the minimum value of (distance from -6) - (distance from 1)

Make a number line with -6 and 1 on it. (-6)..........................(1)

Think of a point in the center of -6 and 1. Its distance from -6 is equal to distance from 1 and hence (distance from -6) - (distance from 1) = 0 .

What if instead, the point x is at -6? Distance from -6 is 0 and distance from 1 is 7 so (distance from -6) - (distance from 1) = 0 - 7 = -7

If you keep moving to the left, (distance from -6) - (distance from 1) will remain -7 so the minimum value is -7.

In the above example, the absolute value function f(x), which is sum of absolute functions, can not be negative for any value of x. Kindly clarify whether the minimum value of f(x) is 7 or -7.

If f(x) = | 1 - x | + | x - 1 |, then minimum value of f(x) is 0 for x = 1. Kindly comment.

I am wondering how can this method be used in questions where there are negative between terms : Ex: minimum value of : |x+6|-|x-1| ?

You can do it with a negative sign too. You want to find the minimum value of (distance from -6) - (distance from 1)

Make a number line with -6 and 1 on it. (-6)..........................(1)

Think of a point in the center of -6 and 1. Its distance from -6 is equal to distance from 1 and hence (distance from -6) - (distance from 1) = 0 .

What if instead, the point x is at -6? Distance from -6 is 0 and distance from 1 is 7 so (distance from -6) - (distance from 1) = 0 - 7 = -7

If you keep moving to the left, (distance from -6) - (distance from 1) will remain -7 so the minimum value is -7.

In the above example, the absolute value function f(x), which is sum of absolute functions, can not be negative for any value of x. Kindly clarify whether the minimum value of f(x) is 7 or -7.

If f(x) = | 1 - x | + | x - 1 |, then minimum value of f(x) is 0 for x = 1. Kindly comment.

Thanks.

Arun.

Sum of two absolute functions cannot be negative but difference can be. The original post discusses the sum of absolute functions. ficklehead asked about f(x) which is difference between two absolute functions. The '-7' is the minimum value of f(x) in case of difference.

f(x) = |a| - |b| can easily be negative e.g. if a = 2 and b = 5
_________________

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

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25 Nov 2013, 03:01

VeritasPrepKarishma wrote:

arunspanda wrote:

VeritasPrepKarishma wrote:

You can do it with a negative sign too. You want to find the minimum value of (distance from -6) - (distance from 1)

Make a number line with -6 and 1 on it. (-6)..........................(1)

Think of a point in the center of -6 and 1. Its distance from -6 is equal to distance from 1 and hence (distance from -6) - (distance from 1) = 0 .

What if instead, the point x is at -6? Distance from -6 is 0 and distance from 1 is 7 so (distance from -6) - (distance from 1) = 0 - 7 = -7

If you keep moving to the left, (distance from -6) - (distance from 1) will remain -7 so the minimum value is -7.

In the above example, the absolute value function f(x), which is sum of absolute functions, can not be negative for any value of x. Kindly clarify whether the minimum value of f(x) is 7 or -7.

If f(x) = | 1 - x | + | x - 1 |, then minimum value of f(x) is 0 for x = 1. Kindly comment.

Thanks.

Arun.

Sum of two absolute functions cannot be negative but difference can be. The original post discusses the sum of absolute functions. ficklehead asked about f(x) which is difference between two absolute functions. The '-7' is the minimum value of f(x) in case of difference.

f(x) = |a| - |b| can easily be negative e.g. if a = 2 and b = 5

The linear sum of absolute value function f(x) = |3x + 1| + |2x-3| + |x - 7| is minimum either at x = -1/3 or x = 3/2.

Rewriting the function f(x) as 3 | x + 1/3 | + 2 | x - 3/2 | + | x - 7 |, we have values of x as -1/3, 3/2 and 7 where constituent absolute functions |3x +1 |, | 2x - 3| and |x-7| are minimum respectively.

Let us write these values of x in ascending order: (the values are repeated based on individual weights or coefficients of absolute value functions)

-1/3, -1/3, -1/3, 3/2, 3/2 and 7

The median value of the above list, -1/3 or 3/2 evaluates f(x) as minimum.

Extending this example further, f'(x) = 4 | x + 1/3 | + 2 | x - 3/2 | + | x - 7 | is minimum for x = -1/3. [ -1/3, -1/3, -1/3, -1/3, 3/2, 3/2 and 7; the median is -1/3] Similarly, f"(x) = 3 | x + 1/3 | + 3 | x - 3/2 | + | x - 7 | is minimum for x = 3/2. [ -1/3, -1/3, -1/3, 3/2, 3/2, 3/2 and 7; the median is 3/2]

Thus we may conclude the median value of x evaluates f(x) as minimum.

Kindly comment.

Arun.

Last edited by arunspanda on 08 Jul 2014, 06:18, edited 1 time in total.

In example : |x - 1| + |x-3| + |x + 1| + |x + 6| .. the posts on the number line are : -6, -1 , 1 and 3. In order to minimize value of this expression to 9, how to chose the x ?

Please correct me if I am wrong.

x is that point on the number line whose sum of distances from -6, -1, 1 and 3 is minimum. So basically there is a person each at points -6, -1, 1 and 3. You need to make them all meet by covering minimum distance. Distance between -6 and 3 is 9 which must be covered by these 2 people to meet. These 2 can meet at any point: -6, -1, 0, 1 or 3 etc they will cover a distance of 9 together. If -1 and 1 have to meet too, they need to cover a distance of 2 together. Say, if person at -1 travels down to 1 and -6 and 3 also meet at 1, the minimum distance covered will be 9+2 = 11 and they will all be able to meet. If they instead meet at -1, the situation will be the same and total distance covered will be 11 again. In fact, they can meet at any point between -1 and 1, the total distance covered will be 11.

To check, put x = 1. you get |x - 1| + |x-3| + |x + 1| + |x + 6| = 11 put x = -1, you get |x - 1| + |x-3| + |x + 1| + |x + 6| = 11 put x = 0, you get |x - 1| + |x-3| + |x + 1| + |x + 6| = 11

Responding to a pm:

Quote:

Just so am clear the minimum value of f(x) for the below Try some other combinations. e.g. f(x) = |x - 1| + |x-3| + |x + 1| + |x + 6| f(x) = |2x - 3| + |4x + 7| etc would be 11 and 13/2 respectively? Am kinda confused over which value x will take.Will it be the total shortest distance covered or the point to which they meet

The 11 and 13/2 that you obtained are the minimum values of the respective functions f(x). This is the total minimum distance covered. The point at which they meet is the value of x i.e. For first question, whenever x is in this range: -1 <= x <= 1, f(x) will take the value 11. For second question, when x = -4/7, f(x) will take the minimum value 13/2.
_________________

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

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14 Feb 2016, 06:41

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).
_________________

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).

Yes, the game changer will be at the transition point.If one or more people stay at home, the length of the trip shortens.

These two posts will help you solidify the concept:

Re: If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II) [#permalink]

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11 Oct 2016, 00:48

here is how solved it>>>

we have=> |4x-1|+|x-3|+|x+1| so to have the minimum value of this function we should have the minimum value of |4x-1|.why? because any value we put of x in all the 3 mods and add all of them...it surely wont be the minimum value.therefore, we should produce a minimum value of mod i.e 0 in the that particular mod which will give us the highest value after substituting the value of x and that mod is |4x-1|.

so, |4x-1|=0 4x-1=0 & -4x+1=0 4x=1 4x=1 x=1/4 x=1/4

remember here it doesnt matter whether we multiply with -1.

put x=1/4 in f(x) we get f(x) = 4

just to be sure that it the minimum value put x=0 and see. we get f(x)=5 therefore, minimum value of f(x)= 4

we applied this method as all the mods are adding up,therefore we needed some negative values to get the minimum value of the addition, but since they are all mods we cant get -ves,so the minimum value of a single mod will be 0.

i dont think most people would understand this approach in one go... but its worth reading another time to understand the process behind it.

Re: If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II) [#permalink]

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12 Oct 2016, 01:20

Here's my way of looking at it, let me know if you see loop holes -

Since we need to minimize the value of f(x), we need to minimize the factor that contributes most to the value of f(x). In this case, it would be 4x. Therefore, making this term 0 should give us the smallest value of f(x). Substituting 1/4 for x gives f(x) to be equal to 4. Option B.