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

Karishma,

So from your method, I infer

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?
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What if the question asks for max value of f(x) ??? disregarding the answer choices ??

Not every function will have a minimum and a maximum value. The greater the value of x, the greater the function will become. It is an infinitely increasing function.
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Many Thanks to all of you for sharing such amazing techniques. I was overwhelmed with mod questions when I started, but your explanations and techniques have helped me build confidence. Bunuel, Karishma, Gurpreet, Shrouded1....Awesome!

How about this approach:

F(x) will be minimum when each individual term in the function has the lowest possible value. So, I get x = 3, -1 and 1/4.
Now, substituting each value of x in F(x), I can easily see that x=1/4 gives me the smallest possible value for F(x) = 4

Thanks,
Rohit

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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The technique is fine but the logic is not sound. Why should we say that the function will take minimum value only when it takes one of these three values? For one of these values, sure one mod will be 0 but the other two could be much greater.
The reason why this works is because the minimum value will be at one of the transition points - the middle point (logic explained in the post on previous page) in case there are odd number of terms OR at two points (and for every value in between) in case there are even number of terms.
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Ok, I get that for f(x) = (4x-1) + (x-3) + (x+1), x must be greater than 1/4, 3 and -1 respectively. But that's where I get lost.

I'm sorry for being so dense on this topic!

Study of the absolute value:

1)take each term into the "| |" and define where it's positive
2)draw a number line with each "edge value" (where each term changes sign)
3)Split the function according to those intervals

Refer to the image

So if x>3 all terms are positive =>f(x) = (4x-1) + (x-3) + (x+1)
if 1/4<x<3 for example you see that 4x-1 is positive, x+1 is positive BUT x-3 is negative => f(x) = (4x-1) + (-)(x-3) + (x+1)

Repeat this operation for each interval and you'll have all possible combinations

Remeber that the each function is valid only in that interval

What I mean is that f(x) = (4x-1) + (-)(x-3) + (x+1) is valid only in the 1/4<x<3 area. Each area has its own function

What I explained above is how to study an abs value from a theoretical point of view, because you original methos is wrong.

Of course this is not required to answer the question, you can try with real number and see what you find out.
But am I trying to explain how an abs function works, for instance your original method

I. f(x) = (4x-1) + (x-3) + (x+1)

II. f(x) = -(4x-1) + -(x-3) + -(x-1)

does not work to find the answer. The function cannot be reduced to that form!

"But why would we bother finding what is positive and what is negative? "
This is required to study an abs value. In my original post you see the graph of F(X), and you notice that is defined into sections.
Each section is one of the intervals above, and in each one of those the fucntion has a different equation.

The concept that I apply here is the same as the one that you would apply to solve
\(y=|x|\)
How would you study this?
for x>0 => y=x
for x<0 => y=-x

Define where the funct is positive, treat each part as a separate equation. The concept in the question is the same, only involves more intervals.

hI Gurpreet,

Why did you change the value of |x-3| to 3-x,

For x >1/4 and x< 3 f(x) is bigger than 4. because f(x) = 4x-1 + 3-x + x+1 = 4x+3 > 3

Kriti

Because when x is less than 3, (x - 3) is negative.
Therefore, |x-3| = - (x - 3) = 3 - x
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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.
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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?

f(x) = -(4x-1)-(x-3)+(x+1)=-4x+5 (calculation mistake above)
f(0) = 5
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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
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Responding to a pm:


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.
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