Find all School-related info fast with the new School-Specific MBA Forum

 It is currently 05 May 2015, 02:44

### 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

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

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# 2 Part question- Max., & Min. value of function!

Author Message
Intern
Joined: 01 Jun 2012
Posts: 7
Followers: 1

Kudos [?]: 3 [1] , given: 0

2 Part question- Max., & Min. value of function! [#permalink]  09 Sep 2012, 06:39
1
KUDOS
1
This post was
BOOKMARKED
Hello Friends!

Try this question...

The function f(x,y) is such that f(x,y) = x^2.y^3 & x+y =25. Where x, & y are non-negative integers. Select one minimum possible, & one maximum possible value for f(x,y). Make only two selections, one in each column.
Attachments

2 Part Qn.JPG [ 26.61 KiB | Viewed 1066 times ]

_________________

Shalabh Jain,
e-GMAT Instructor

Last edited by ShalabhsQuants on 09 Sep 2012, 19:01, edited 2 times in total.
Director
Joined: 22 Mar 2011
Posts: 612
WE: Science (Education)
Followers: 78

Kudos [?]: 591 [0], given: 43

Re: 2 Part question- Max., & Min. value of function! [#permalink]  09 Sep 2012, 10:36
ShalabhsQuants wrote:
Hello Friends!

Try this question...

I checked the OG13 and QuantReview: they don't use neither the term whole number, nor that of natural number.
I guess, in order to avoid the lack of consensus regarding 0. And in fact, why have so many different definitions?
It is much easier to have just integers, which are positive, negative or zero. Further, you can have non-negative integers if you want to include 0,...

Oh, and I don't think the symbol for infinity should be known on the GMAT.
I would say, not a must to solve such questions for a future test taker.
_________________

PhD in Applied Mathematics
Love GMAT Quant questions and running.

Intern
Joined: 01 Jun 2012
Posts: 7
Followers: 1

Kudos [?]: 3 [0], given: 0

Re: 2 Part question- Max., & Min. value of function! [#permalink]  09 Sep 2012, 18:43
EvaJager wrote:
ShalabhsQuants wrote:
Hello Friends!

Try this question...

I checked the OG13 and QuantReview: they don't use neither the term whole number, nor that of natural number.
I guess, in order to avoid the lack of consensus regarding 0. And in fact, why have so many different definitions?
It is much easier to have just integers, which are positive, negative or zero. Further, you can have non-negative integers if you want to include 0,...

Oh, and I don't think the symbol for infinity should be known on the GMAT.
I would say, not a must to solve such questions for a future test taker.

Thanks EvaJager. I edit it.
_________________

Shalabh Jain,
e-GMAT Instructor

Senior Manager
Joined: 15 Jun 2010
Posts: 363
Schools: IE'14, ISB'14, Kellogg'15
WE 1: 7 Yrs in Automobile (Commercial Vehicle industry)
Followers: 10

Kudos [?]: 205 [0], given: 50

Re: 2 Part question- Max., & Min. value of function! [#permalink]  09 Sep 2012, 19:04
Min value of f(x,y) = 0, If we choose either x or y as 0 and the other as 25. The product x^2y^3 becomes 0.
Max value of f(x,y) = Infinity. Since x+y =25, and both are integers, we can choose y as a +ve number such as 10000xxxxxx000 and x as -ve no such as 999999xxxxx75. If we add we will get 25, but if we square x it will be a positive expression and the value of f(x,y) can go up to infinity.
_________________

Regards
SD
-----------------------------
Press Kudos if you like my post.
Debrief 610-540-580-710(Long Journey): from-600-540-580-710-finally-achieved-in-4th-attempt-142456.html

Intern
Joined: 01 Jun 2012
Posts: 7
Followers: 1

Kudos [?]: 3 [0], given: 0

Re: 2 Part question- Max., & Min. value of function! [#permalink]  09 Sep 2012, 21:25
SOURH7WK wrote:
Min value of f(x,y) = 0, If we choose either x or y as 0 and the other as 25. The product x^2y^3 becomes 0.
Max value of f(x,y) = Infinity. Since x+y =25, and both are integers, we can choose y as a +ve number such as 10000xxxxxx000 and x as -ve no such as 999999xxxxx75. If we add we will get 25, but if we square x it will be a positive expression and the value of f(x,y) can go up to infinity.

As far as Minimum value of f(x,y) is concerned, your answer is correct. But Maximum value is not infinity. Remember x, & y both are non-negative integers. Minimum values of x, & y would be 0 only, whereas maximum as 25.
_________________

Shalabh Jain,
e-GMAT Instructor

Senior Manager
Joined: 15 Jun 2010
Posts: 363
Schools: IE'14, ISB'14, Kellogg'15
WE 1: 7 Yrs in Automobile (Commercial Vehicle industry)
Followers: 10

Kudos [?]: 205 [0], given: 50

Re: 2 Part question- Max., & Min. value of function! [#permalink]  09 Sep 2012, 23:04
ShalabhsQuants wrote:
SOURH7WK wrote:
Min value of f(x,y) = 0, If we choose either x or y as 0 and the other as 25. The product x^2y^3 becomes 0.
Max value of f(x,y) = Infinity. Since x+y =25, and both are integers, we can choose y as a +ve number such as 10000xxxxxx000 and x as -ve no such as 999999xxxxx75. If we add we will get 25, but if we square x it will be a positive expression and the value of f(x,y) can go up to infinity.

As far as Minimum value of f(x,y) is concerned, your answer is correct. But Maximum value is not infinity. Remember x, & y both are non-negative integers. Minimum values of x, & y would be 0 only, whereas maximum as 25.

Ok, I missed that "non" part in the stem. Now looking at the choices (i,e ending with zeros) I tried few combination and got 10^2x15^3 = 33750. So is Option C is the maximum value?
_________________

Regards
SD
-----------------------------
Press Kudos if you like my post.
Debrief 610-540-580-710(Long Journey): from-600-540-580-710-finally-achieved-in-4th-attempt-142456.html

Intern
Joined: 01 Jun 2012
Posts: 7
Followers: 1

Kudos [?]: 3 [0], given: 0

Re: 2 Part question- Max., & Min. value of function! [#permalink]  10 Sep 2012, 00:54
SOURH7WK wrote:
ShalabhsQuants wrote:
SOURH7WK wrote:
Min value of f(x,y) = 0, If we choose either x or y as 0 and the other as 25. The product x^2y^3 becomes 0.
Max value of f(x,y) = Infinity. Since x+y =25, and both are integers, we can choose y as a +ve number such as 10000xxxxxx000 and x as -ve no such as 999999xxxxx75. If we add we will get 25, but if we square x it will be a positive expression and the value of f(x,y) can go up to infinity.

As far as Minimum value of f(x,y) is concerned, your answer is correct. But Maximum value is not infinity. Remember x, & y both are non-negative integers. Minimum values of x, & y would be 0 only, whereas maximum as 25.

Ok, I missed that "non" part in the stem. Now looking at the choices (i,e ending with zeros) I tried few combination and got 10^2x15^3 = 33750. So is Option C is the maximum value?

You are right!
Let's understand the concept. This will eliminate the hit & trial approach for these kinds of questions.

If a function is such that f(a,b)=a^m.b^n, & a+b= constant, then f(a,b) would be maximum when a/m=b/n.

Coming to this question....

Given is x+y=25 =constant. for f(x,y)=x^2.y^3 to be max., x/2 should be equal to y/3 =>x/2=y/3 or x=2y/3.

By plugging in this value in x+y=25, we get 2y/3+y=25 => y=15, & x=10.

So Maximum of f(x,y)= 10^2.15^3 = 337500.
_________________

Shalabh Jain,
e-GMAT Instructor

Senior Manager
Joined: 15 Jun 2010
Posts: 363
Schools: IE'14, ISB'14, Kellogg'15
WE 1: 7 Yrs in Automobile (Commercial Vehicle industry)
Followers: 10

Kudos [?]: 205 [0], given: 50

Re: 2 Part question- Max., & Min. value of function! [#permalink]  10 Sep 2012, 01:47
ShalabhsQuants wrote:

If a function is such that f(a,b)=a^m.b^n, & a+b= constant, then f(a,b) would be maximum when a/m=b/n.

Coming to this question....

Given is x+y=25 =constant. for f(x,y)=x^2.y^3 to be max., x/2 should be equal to y/3 =>x/2=y/3 or x=2y/3.

By plugging in this value in x+y=25, we get 2y/3+y=25 => y=15, & x=10.

So Maximum of f(x,y)= 10^2.15^3 = 337500.

How you have derived that formula. The formula seems very conditional with a+b constant & only for maximum value. Is there any partial derivative involved??
_________________

Regards
SD
-----------------------------
Press Kudos if you like my post.
Debrief 610-540-580-710(Long Journey): from-600-540-580-710-finally-achieved-in-4th-attempt-142456.html

Director
Joined: 22 Mar 2011
Posts: 612
WE: Science (Education)
Followers: 78

Kudos [?]: 591 [1] , given: 43

Re: 2 Part question- Max., & Min. value of function! [#permalink]  10 Sep 2012, 04:13
1
KUDOS
SOURH7WK wrote:
ShalabhsQuants wrote:

If a function is such that f(a,b)=a^m.b^n, & a+b= constant, then f(a,b) would be maximum when a/m=b/n.

Coming to this question....

Given is x+y=25 =constant. for f(x,y)=x^2.y^3 to be max., x/2 should be equal to y/3 =>x/2=y/3 or x=2y/3.

By plugging in this value in x+y=25, we get 2y/3+y=25 => y=15, & x=10.

So Maximum of f(x,y)= 10^2.15^3 = 337500.

How you have derived that formula. The formula seems very conditional with a+b constant & only for maximum value. Is there any partial derivative involved??

Formally, yes, it is by partial derivatives, looking for extremum point...

A sort of justification without partial derivatives:
For any real numbers $$x$$ and $$y$$, $$\, (x + y)^2 \geq{4xy}$$. Equality holds if and only if $$x=y$$ (the given inequality is equivalent to $$(x-y)^2\geq{0}$$. In words: when the sum of two real numbers is constant, the maximum product of the two numbers is obtained when they are each equal to half of the sum.

In our case, the sum is constant, but in the product we have two different powers, 2 and 3. Intuitively, the maximum will be obtained for a weighted average between $$x$$ and $$y$$, $$y$$ being closer to 25 as in the product it has a greater power, but still not "too far away" from the half of the sum.

But, since here we have integers and in addition, it is a GMAT multiple choice question, we can use some number properties.
The possible answers (after we eliminate infinity) are all multiples of 5, and since the sum$$x+y=25$$ is a multiple of 5, if one of the numbers is multiple of 5, then the other one is also. And of course, $$y$$ should be greater than $$x.$$
Therefore, we only have to check $$x=5, \, y=20$$ and $$x=10, \, y=15.$$
_________________

PhD in Applied Mathematics
Love GMAT Quant questions and running.

Intern
Joined: 17 Aug 2012
Posts: 3
Followers: 0

Kudos [?]: 1 [0], given: 4

Re: 2 Part question- Max., & Min. value of function! [#permalink]  17 Sep 2012, 17:18
Thanks for sharing this question, but I think it's too mathematical to appear in a real exam.
Re: 2 Part question- Max., & Min. value of function!   [#permalink] 17 Sep 2012, 17:18
Similar topics Replies Last post
Similar
Topics:
5 Another fresh question on 2 Part- Quadratic function 5 25 Jan 2013, 03:56
Ratio, Min/max problem 4 17 Mar 2011, 18:38
21 In a certain company, the formula for maximizing profits is 15 09 Feb 2009, 02:12
DS - min and max 14 05 Jul 2006, 15:55
ds - min/max 15 20 Oct 2005, 08:54
Display posts from previous: Sort by