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Karishma, Is there a way to solve this by setting boiundary conditions. ie. x<-1,-1<x<1/4,1/4<x<3,x>3?? I tried but wasnt able to make sense

VeritasPrepKarishma wrote:

Q. If f(x) = |4x - 1| + |x-3| + |x + 1|, what is the minimum value of f(x)?

(A) 3 (B) 4 (C) 5 (D) 21/4 (E) 7

(Still high on mods! Next week, will make questions on some other topic.)

If you mean whether you can make equations using positive and negative values, you cant do that with minimum/maximum questions. You don't really have a value to equate them to. e.g. |4x - 1| + |x-3| + |x + 1| = 10 is workable but minimum value of f(x) isn't. You will need to find the value at the critical points and then figure how f(x) changes or just use the number line.
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Can you please explain the example which has negative between the modulus (could not understand fully the explaination given by you earlier). Also if in a question we have a combination of + and - then how to go about it?

Can you please explain the example which has negative between the modulus (could not understand fully the explaination given by you earlier). Also if in a question we have a combination of + and - then how to go about it?

When you add two mods, you try to add up the distances e.g. |x - 1| + |x - 5| = 10 you try to find the point where distance from 1 and distance from 5 adds up to give you 10.

When you subtract two mods, you subtract out the distances e.g. |x - 1| - |x - 5| = 3 you try to find the point where distance from 1 and distance from 5 have a difference of 3. You know that at x = 3, distances from 1 and from 5 are equal (distance of 1 from 3 is 2 and distance of 5 from 3 is also 2). At x = 4, the difference between the distances will be 2. At x = 4.5, the difference between the distances will be 3.

The subtraction is a little less intuitive and will take more practice. Questions with both + and - would be too complicated though do-able. Most people will probably not get any mods question with more than 2 terms.
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I love this question. One must understand that f(x) = |4x-1| + |x-3| + |x+1| means the sum of the distances of x to 1/4, 3 and -1. The best way to minimize is to zero out the distance in the middle.

==========(-1)==========(0)======(1/4)=========(3)======= So if x = 1/4 |4x-1| = 0 |x+1| = 1 1/4 |x-3| = 2 3/4

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

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 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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I am wondering why we cannot take the positive and negative cases of f(x) = |4x - 1| + |x-3| + |x + 1| and solve for x that way?

In other words, f(x) = |4x - 1| + |x-3| + |x + 1|

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

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

Thanks!

It's not that easy!

if you wanna study the absolute value, more math is required. You have to study each \(abs>0\) so \(4x-1>0\) and \(x-3>0\) and \(x+1>0\) \(x>\frac{1}{4}\) and \(x>3\) and \(x>-1\) so 4x-1 is positive for x>1/4, x-3 is +ve for x>3 and x-1 is +ve for x>-1

Now you have to split the original function into the areas defined above: \(x<-1\) all functions are negative \(f(x) = -(4x-1) + -(x-3) + -(x-1)\) if \(-1<x<\frac{1}{4}\) the third term is positive,the others negative \(f(x) = -(4x-1) + -(x-3) +(x-1)\) and so on...

You cannot take all positive or all negative, you have to study each function in all possible intervals Below there is the graph of F(x) that I hope will make thing clear. As you see there are 4 functions, each one defined in the intervals above, so your way of studying the abs value (reducing all to 2 functions) is incomplete

Hope it's clear

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