M28-10

The trick with these questions that ask you about max/min of a second degree polynomial is that you will invariably get a value of the form (a\(\pm\)b) wherein, the minimum value of any square = 0. Thus try to create perfect squares out of the given polynomials.


Realize that \((a\pm b)^2 = a^2 \pm 2*a*b + b^2\).

Given expression: \(−3x^2+12x−2y^2−12y−39\) ---> try to create perfect squares :

\((-3x^2+12x) + (-2y^2-12y)-39 ---> -3 (x^2-4x) -2 (y^2+6y) -39 ----> 3 (x^2-4x+4-4) -2 (y^2+6y+9-9) -39\)


---> \(3 (x^2-4x+4-4) -2 (y^2+6y+9-9) -39\) ---> \(-3[(x-2)^2-4] -2 [(y+3)^2-9]-39\) ----> \(-3(x-2)^2+12 -2(y+3)^2+18-39\)

----> \(-3(x-2)^2+12 -2(y+3)^2+18-39\) --->\(-3(x-2)^2-2(y+3)^2-9\)

As mentioned above, the minimum value of a perfect square = 0 --> to maximize the given expression,

you need to put \((x-2)^2 =0\) and\((y+3)^2=0\) ---> giving you 0+0-9=-9 as the maximum value.

Any other values of\((x-3)^2\) and \((y+3)^2\)will give you a smaller negative value hence 'minimizing' instead of maximizing the expression.

Hope this helps.

If you are familiar with basic differentiation, it would be very quick.

Solution: Let the given equation be 1.

Step 1.Find differentiation of equ 1 wr.to "X" and equate to "0" (This is the standard rule for a differentiation in order to get maximum value)
Hence, -6x+12=0
therefore x=2

Step 2.Then differentiate the equ 1 wr.to "Y" and equate to "0"
-4y-12=0
therefore y=-3

Step 3.Submit both x=2 and y=-3 in the equ. 1 to get the maximum value.
Therefore the answer is -9.

Differentiation approach can save lot of time while determining the maximum value of an expression.

i solved this question in following way:
to maximize this expression, we need to maximize terms containing 'x', i.e
\(−3x^2+12x\)
and we need to maximize terms containing 'y', i.e
\(−2y^2−12y\)
So,
\(−3x^2+12x\) will be maximum when x is positive and \(x=2\).
\(−2y^2−12y\) will be maximum when y is negative and \(y=-3\)
so by plugging in \(x=2\) and \(y=-3\) in given expression we get \(-9\) and that's the maximum value we can get from this expression.

Not sure if this approach helps but I kinda took the differential approach and found the answer in a minute...

d/dx(expression)= 0 to find the value of x and d/dy(expression)=0 to find the value of y. This is the way we determine maximas in differentiation. This is a basic second order differential and anyone who has been through grade 12 in science/Math/Physics should be able to solve this.

d/dx=-6x+12=0, x=2

Similarly, d/dy= -4y-12=0, y=-3,

Sub x and y in the equation and you get maximum value of the expression as -9.

Hope this helps and please give me a kudos if you relate to the approach!

I found out why [b]Bunuel breaks out the "-39", and have found a formula to find the maximum of a second degree polynomial. A detailed understanding and step-by-step walkthrough is included below.[/b]

Disclaimer: This post is "wordy"; however, as any of us have likely determined throughout our studying, understanding the entire thought process, or logic, of how to arrive at a given answer is important.

Logic
The key to this question is to recognize exactly what it is testing, which is: "how do you find the maximum of a second degree polynomial?"

Upon recognizing this "task" that the question is testing, deciphering Bunuel's logic of breaking out the "-39" to multiple terms makes complete sense.

As background, to find the maximum of a second degree polynomial (ax^2 + bx + c), the formula is: "(c - ((b^2) / (4a)))". More on that here: https://study.com/academy/lesson/find-the-maximum-value-of-a-function-lesson-practice-quiz.html. As a further note of background, if the 'a-coefficient' is positive, the polynomial will have a minimum, and if the 'a-coefficient' is negative, the polynomial will have a maximum. Regardless, the formula you use to find either maximum or minimum is the exact same: (c - ((b^2) / (4a))).

Upon recognizing what the test is asking/testing, and knowing the formula stated above to find the maximum of a second degree polynomial, you can see why Bunuel breaks out the "-39" to create two separate second degree polynomials; as a way to create two polynomials that you can then find the respective maximum values for.

Step-by-step
1.) -3x^2 + 12x - 2y^2 - 12y - 39 --> you should be able to recognize that there is a "complete" 2nd degree polynomial in "-2y^2 - 12y - 39"; however, the "-3x^2 + 12x" is "incomplete" as there is no "c" value.
2.) At this point, you need to ask yourself, "How can I make two complete 2nd degree polynomials?" Specifically, how can you write "-3x^2 + 12x +/- R" and "-2y^2 - 12y +/- S"?, R and S would be derived from -39.
Note: "R" and "S" are term names that I made up to help distinguish in this post between the respective polynomials, as opposed to simply calling them both "c", which is technically their term name for a polynomial.
2a.) To find values for R and S, you need to ask yourself, "what two numbers that add up to -39 (either completely or with values left over) can I use that would 'complete' X's and Y's respective polynomials, and result in each polynomial to have the a-coefficient factored out?" For example, the "R" (or "c") value of the "-3x^2 + 12 +/- R" polynomial would have to be divisible by 3, since "-3" is the a-coefficient, in order to create ease of factoring. Similarly, the "S" (or "c") value of the "-2y^2 - 12y +/- S" polynomial would have to be divisible by 2, for the exact same reason.
2b.) If you're following up to this point, there are multiple ways to break out "-39" to be numbers that would "complete" each polynomial (e.g., (-12)+(-18)+(-9)=-39; (-3)+(-3)+(-33)=-39; etc.). Essentially, any of these numbers will work to aid you in arriving at the answer, just remember that the R and S terms should be divisible by their respective a-coefficient.
3.) For the sake of showing why/how Bunuel broke out -39 to -12, -18, and -9, we will do that here.
3a.) The new formula from the prompt is: -3x^2 + 12x - 12 - 2y^2 - 12y - 18 - 9 --> Do you see how you now have two "complete" polynomials" for X and Y with -9 left over?
3b.) Mark the polynomials with parentheses to distinguish between them... (-3x^2 + 12x - 12) + (-2y^2 - 12y - 18) - 9
Steps 4 and 5 are "new" compared to the previous posts I have seen on this question.
4.) Remember how I mentioned above how to find the maximum of a polynomial (c - ((b^2) / 4a)? Use this formula to find the maximum value for each newly created "complete" polynomial
4a.) For the X-polynomial, the maximum formula becomes: (-12 - ((12^2) / 4(-3))) --> (-12 - ((144) / (-12))) --> (-12 - (-12)) --> (-12 + 12) --> 0... the maximum value for the X-polynomial is "0"... hold on to that thought
4b.) For the Y-polynomial, the maximum formula becomes: (-18 - ((-12^2) / 4(-2))) --> (-18 - ((144) / (-8))) --> (-18 - (-18)) --> (-18 + 18) --> 0... the maximum value for the Y-polynomial is "0"... again, hold on to that though; we aren't done!
5.) Now, combine all of the new numbers. The new equation finally breaks down to: 0 + 0 - 9... which finally results in: -9 (answer B).

As always, please feel free to share thoughts, questions, comments, and/or concerns. As you can likely understand, this question threw me for a while, and I did not understand Bunuel's logic of breaking out the "-39", thus I researched how to find the max of a second degree polynomial, did not see the affiliated logic shared throughout the thread, and felt compelled to share. Hope this helps!

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

I think this is a poor-quality question and I agree with explanation. This question gives me a vibe of the Indian MBA entrance exam i.e. CAT. I don't think this type of question will ever be asked in GMAT (I could be wrong though), but it is a good tool to have in your Quant armoury.

I think this is a high-quality question and I agree with explanation.