The function f(x,y) is such that f(x,y) = x^2.y^3 & x+y =25. Where x,

Question

The function f(x,y) is such that  f(x,y) = x^2y^3  & x + y = 25. Where x and 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.

ShalabhsQuants · Accepted Answer

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.

EvaJager · Answer

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.

SOURH7WK · Answer

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.

ShalabhsQuants · Answer

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.

SOURH7WK · Answer

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?

SOURH7WK · Answer

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??

EvaJager · Answer

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.

ManifestDreamMBA · Answer

This is interesting concept. On further research, turns out this has been derived from weighted AM-GM concepts.

sitabjab · Answer

it is possible to solve the problem by basic maximum minimum problem using differentiation.

cheshire · Answer

this problem  can also be solved using basic calculus techniques  (i.e.,  optimization with derivatives ), even though the final answers need to be integers. However, on the GMAT derivatives are not tested.

Vivek1707 · Answer

One quick way to solve this quesiton

Since we are given that x+y = 25
We note that x and y are non negetive hence take values of x and y as 0 and 25 ----> gives you F(x^2y^3) = 0 hence this ll be the minimum value.

Now, for the max value notice
+ Value cannot be infinity since there are fixed number of values that can be tested thought the equation x+y = 25
+ Other answers have a units digit of 0
Hence test values such as 10 and 15 which will give you are units digit of 9. Since y is cubed it ll result in a higher number, try 15 for y you ll get 10^2*15^3 which is exactly 337,500.

shashi1277 · Answer

Given,
f(x,y)=x^2y^3
x + y = 25
x,y≥0 integers
 
For Minimum,
x=0, y=25 → f=0
x=25, y=0 → f=0
So minimum value of f(x,y) is 0
 
For Maximum,
y=25−x
f(x)=x^2(25−x)^3
So the value of f(x,y) will be maximised if the value of y is greater than x,
and it could be near the middle but not exactly at middle and neither at extreme end,
example for x=12, y=13, and x=1, y=24, still value of f(x,y) will be not maximum
 
So logic would be to consider the power ratio of both.
variable with the higher exponent contributes more to the product
 
consider a scale of 25 units (x+y=25)
Now, power ratio from f(x,y) of x:y is 2:3
the part of y out of 25 will be 3/(2+3) * 25 = 15 units
and for x will be 2/5 * 25 = 10 units
Plug these in the f(x,y)= 10^2 * 15^3 = 100*3375 = 337500
So maximum value of f(x,y) is 337500

egmat · Answer

Given: 
f(x,y) = x2y3
x + y =  25 
x, y are  non-negative integers  (meaning  0, 1, 2, ... 25 )

Step 1: Finding Minimum 
 Key Insight:  Since f(x,y) = x2y3, if  either  x =  0  OR y =  0 , the entire product becomes  0 .

When x =  0 : y =  25 , so f( 0 ,  25 ) =  0 2 ×  25 3 =  0 
When x =  25 : y =  0 , so f( 25 ,  0 ) =  25 2 ×  0 3 =  0

Minimum = 0

Step 2: Finding Maximum 
  Optimization Principle:   For expressions like x2y3 with constraint x + y = constant, the maximum occurs when resources are allocated in the  ratio of the exponents .

Here: exponent of x is  2 , exponent of y is  3 
 Therefore,  optimal ratio is x : y =  2 : 3

With x + y =  25  and ratio  2:3 :
x =  25  × ( 2 / 5 ) =  10 
y =  25  × ( 3 / 5 ) =  15

Calculating: 
f( 10 ,  15 ) =  10 2 ×  15 3 =  100  ×  3375  =  337,500

Verification with neighboring values: 
f( 9 ,  16 ) =  81  ×  4096  =  331,776  (less than  337,500 )
f( 11 ,  14 ) =  121  ×  2744  =  332,024  (less than  337,500 )

Maximum = 337,500

Answer: Minimum = 0, Maximum = 337,500

dp1234 · Answer

egmat

Found that that you have arrived at optimization principle using the calculus. The process would take a lot of time and also calculus is not part of gmat scope. So i think best way to arrive at Max value is by eliminating options. OR is there any generalist optimization priniciple/logic ? IMO remembering so many specific cases would not be practical.

egmat · Answer

You can absolutely solve this by elimination. Let me walk you through how.
First,  infinity ?  Not possible.  x and y are just numbers between  0  and  25 . You'll always get a finite answer. Gone.
That leaves  29,600 ,  337,500 , and  400,000 . Now we need to figure out which one is the maximum.

If you don't know any optimization principle, what would you naturally try? One of two things:
 "y is being cubed, that's powerful, let me give almost everything to y."

Okay, let's try that. And this is very easy to compute:

x =  1 , y =  24 : f =  1  ×  13,824  =  13,824

x2 became  1 , and  1  times anything is just... that thing. We basically threw x away. That clearly didn't work.

"Okay then, let me just split it equally."

x =  12 , y =  13 : f =  144  ×  2,197  =  316,368

Now that's way better! We're already close to  337,500 .

So look at what we just learned from two tries. Giving almost everything to y was a disaster because x died. Splitting equally was pretty strong because both contributed. But y is still the more powerful one (cubed vs squared), so maybe y deserves a  little more  than equal?

From 12/13, slide a little toward y. And go for clean numbers that are easy to compute:

x =  10 , y =  15 : f =  100  ×  3,375  =  337,500  ✓ that matches an option!

Check one neighbor to confirm:

x =  9 , y =  16 : f =  81  ×  4,096  =  331,776  ← went down

We're at the peak.  400,000  is unreachable.

✓   Maximum = 337,500

But Why Did 10 and 15 Turn Out to Be the Sweet Spot? 
No formula needed. It's pure logic.

Look at what x2y3 actually means when you expand it:

x2 × y3 =  x × x  ×  y × y × y

That's x showing up  2  times and y showing up  3  times.  5  appearances total.

You have  25  to split. If you spread it fairly across all  5  appearances, each one gets  25  ÷  5  =  5 .

x shows up  2  times → x =  2  ×  5  =  10  
 y shows up  3  times → y =  3  ×  5  =  15

That's it. y gets more not because of some optimization formula, but because y literally shows up more times in the multiplication. It's just fair distribution.

Answer: Minimum = 0, Maximum = 337,500

Adit_ · Answer

or the second part you only need to check like couple values.
For maximum part, during multiplications, you get the highest product when values are close together
Like 12+13
or 11+14 and so on
and 15+10

Actually one more value to show that the next value when chosen will make the value go down.
So 4 substitutions ideally. And we see that that 10+15 combo indeed gives us the max value hence results in the answer.

minimum f(x,y)	maximum f(x,y)
		0
		29600
		337500
		400000
		infinity

The function f(x,y) is such that f(x,y) = x^2.y^3 & x+y =25. Where x,

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The function f(x,y) is such that f(x,y) = x^2.y^3 & x+y =25. Where x,

Prep Toolkit

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