GMAT Question of the Day - Daily to your Mailbox; hard ones only

It is currently 19 Oct 2018, 21:21

Close

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel

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

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Veritas Prep GMAT Instructor
User avatar
P
Joined: 16 Oct 2010
Posts: 8397
Location: Pune, India
Re: Question of the Day - II  [#permalink]

Show Tags

New post 06 Aug 2013, 22:52
vaishnogmat wrote:
VeritasPrepKarishma wrote:
eaakbari wrote:
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?




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.
_________________

Karishma
Veritas Prep GMAT Instructor

Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >

GMAT self-study has never been more personalized or more fun. Try ORION Free!

Manager
Manager
avatar
Joined: 12 Feb 2012
Posts: 125
Re: If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II)  [#permalink]

Show Tags

New post 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?
Veritas Prep GMAT Instructor
User avatar
P
Joined: 16 Oct 2010
Posts: 8397
Location: Pune, India
Re: If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II)  [#permalink]

Show Tags

New post 10 Sep 2013, 21:45
alphabeta1234 wrote:
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
_________________

Karishma
Veritas Prep GMAT Instructor

Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >

GMAT self-study has never been more personalized or more fun. Try ORION Free!

Manager
Manager
avatar
Joined: 04 Oct 2013
Posts: 154
Location: India
GMAT Date: 05-23-2015
GPA: 3.45
GMAT ToolKit User Premium Member Reviews Badge
Re: Question of the Day - II  [#permalink]

Show Tags

New post 24 Nov 2013, 07:16
VeritasPrepKarishma wrote:
ficklehead wrote:
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.
Veritas Prep GMAT Instructor
User avatar
P
Joined: 16 Oct 2010
Posts: 8397
Location: Pune, India
Re: Question of the Day - II  [#permalink]

Show Tags

New post 24 Nov 2013, 20:20
arunspanda wrote:
VeritasPrepKarishma wrote:
ficklehead wrote:
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
_________________

Karishma
Veritas Prep GMAT Instructor

Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >

GMAT self-study has never been more personalized or more fun. Try ORION Free!

Veritas Prep GMAT Instructor
User avatar
P
Joined: 16 Oct 2010
Posts: 8397
Location: Pune, India
Re: Question of the Day - II  [#permalink]

Show Tags

New post 01 Dec 2013, 23:49
1
VeritasPrepKarishma wrote:
ficklehead wrote:
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.
_________________

Karishma
Veritas Prep GMAT Instructor

Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >

GMAT self-study has never been more personalized or more fun. Try ORION Free!

Manager
Manager
User avatar
Status: folding sleeves up
Joined: 26 Apr 2013
Posts: 139
Location: India
Concentration: Finance, Strategy
GMAT 1: 530 Q39 V23
GMAT 2: 560 Q42 V26
GPA: 3.5
WE: Consulting (Computer Hardware)
Re: If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II)  [#permalink]

Show Tags

New post 25 Mar 2014, 14:06
VeritasPrepKarishma wrote:
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.)



Here is the easiest solution
Attachments

image.jpg
image.jpg [ 545.89 KiB | Viewed 3527 times ]

Current Student
User avatar
Joined: 12 Aug 2015
Posts: 287
Concentration: General Management, Operations
GMAT 1: 640 Q40 V37
GMAT 2: 650 Q43 V36
GMAT 3: 600 Q47 V27
GPA: 3.3
WE: Management Consulting (Consulting)
If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II)  [#permalink]

Show Tags

New post 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).
_________________

KUDO me plenty

Veritas Prep GMAT Instructor
User avatar
P
Joined: 16 Oct 2010
Posts: 8397
Location: Pune, India
Re: If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II)  [#permalink]

Show Tags

New post 16 Feb 2016, 23:20
1
shasadou wrote:
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:

http://www.veritasprep.com/blog/2011/01 ... edore-did/
http://www.veritasprep.com/blog/2011/01 ... s-part-ii/
_________________

Karishma
Veritas Prep GMAT Instructor

Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >

GMAT self-study has never been more personalized or more fun. Try ORION Free!

Intern
Intern
avatar
Joined: 09 May 2016
Posts: 10
Re: If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II)  [#permalink]

Show Tags

New post 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.
Intern
Intern
avatar
Joined: 12 Sep 2016
Posts: 1
Re: If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II)  [#permalink]

Show Tags

New post 04 Dec 2016, 19:05
Don't know if this method is already discussed in the forum in the above posts:

F(x) has three different expressions inside mods. None of them can be negative if the mods are removed. Hence, the three mods can take the value of 'zero' to be closest towards the minimum value. This gives us three different values of x = 1/4,3,-1. Putting, three values in the expression, once at a time, yields the minimum value of f(x) when x = 1/4. Hence, F(x) = 4
Senior CR Moderator
User avatar
V
Status: Long way to go!
Joined: 10 Oct 2016
Posts: 1381
Location: Viet Nam
GMAT ToolKit User Premium Member
Re: If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II)  [#permalink]

Show Tags

New post 04 Dec 2016, 22:52
VeritasPrepKarishma wrote:
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


Apply two inequilities below
(1) For all \(x,y \in R\) we have \(|x|+|y| \geq |x+y|\). Sign "=" occurs \(\iff xy \geq 0\)
(2) For all \(x \in R\) we have \(|x| \geq 0\)

Now, let's apply these inequilities into \(f(x)\)
\(f(x)=|4x - 1| + |x-3| + |x + 1|=|4x-1|+(|3-x|+|x+1|) \geq |4x-1|+|3-x+x+1|=|4x-1|+4 \geq 0+4=4\).

\(min f(x)=4 \iff \Bigg\{\begin{split} 4x-1=0 \\ (x+1)(3-x) \geq 0 \end{split}
\iff \Bigg\{\begin{split} x=\frac{1}{4} \\ -1 \leq x \leq 3 \end{split} \implies x=\frac{1}{4}\)
_________________

Actual LSAT CR bank by Broall

How to solve quadratic equations - Factor quadratic equations
Factor table with sign: The useful tool to solve polynomial inequalities
Applying AM-GM inequality into finding extreme/absolute value

New Error Log with Timer

Non-Human User
User avatar
Joined: 09 Sep 2013
Posts: 8459
Premium Member
Re: If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II)  [#permalink]

Show Tags

New post 29 Mar 2018, 08:52
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

GMAT Books | GMAT Club Tests | Best Prices on GMAT Courses | GMAT Mobile App | Math Resources | Verbal Resources

GMAT Club Bot
Re: If f(x)= |4x - 1| + |x-3| + |x + 1| (Question of the Day-II) &nbs [#permalink] 29 Mar 2018, 08:52

Go to page   Previous    1   2   3   [ 53 posts ] 

Display posts from previous: Sort by

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

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  


Copyright

GMAT Club MBA Forum Home| About| Terms and Conditions and Privacy Policy| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.