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Min/Max question

by Mo2men » Mon Feb 20, 2017 2:33 am
What is the maximum value of −3x^2+12x−2y^2−12y−39?

A. −39
B. −9
C. 0
D. 9
E. 39

OA:B
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by GMATGuruNY » Mon Feb 20, 2017 3:11 am
Mo2men wrote:What is the maximum value of −3x^2+12x−2y^2−12y−39?

A. −39
B. −9
C. 0
D. 9
E. 39
This type of question is typically solved by COMPLETING THE SQUARE -- a process that is beyond the scope of the GMAT.
Feel free to ignore this problem.

A GMAT-friendly approach:

Maximize the value of −3x² + 12x:
If x=0, then −3x^2 + 12x = 0.
If x=1, then −3x^2 + 12x = 9.
If x=2, then −3x^2 + 12x = 12.
If x=3, then −3x^2 + 12x = 9.
The results above indicate that the greatest possible value of −3x²+12x is 12.

Maximize the value of −2y² − 12y = -2(y² + 12y).
Here, the value will be maximized if the expression in red is NEGATIVE.
Test negative values in −2y²−12y.
If y=-1, then −2y² − 12y = 10.
If y=-2, then −2y² − 12y = 16.
If y=-3, then −2y² − 12y = 18.
If y=-4, then −2y² − 12y = 16.
The results above indicate that the greatest possible value of −2y²−12y is 18.

Thus:
The greatest possible value of −3x²+12x−2y²−12y−39 = (greatest possible value of −3x²+12x) + (greatest possible value of −2y²−12y) − 39 = 12 + 18 - 39 = -9.

The correct answer is B.
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by Brent@GMATPrepNow » Mon Feb 20, 2017 7:52 am
Mo2men wrote:What is the maximum value of −3x² + 12x − 2y² − 12y − 39?

A. −39
B. −9
C. 0
D. 9
E. 39
Here's the COMPLETING THE SQUARE approach (which you do NOT need to know for the GMAT):

−3x² + 12x − 2y² − 12y − 39 = −3(x² - 4x) − 2(y² + 6y) − 39
= −3(x² - 4x + 4 - 4) − 2(y² + 6y + 9 - 9) − 39
= −3(x² - 4x + 4) + 12 − 2(y² + 6y + 9) + 18 − 39
= −3(x - 2)² − 2(y + 3)² − 9
Since (x - 2)² and (y + 3)² are both being multiplied by NEGATIVE VALUES (-3 and -2), we will MAXIMIZE the value of the expression by MINIMIZING the values of (x - 2)² and (y + 3)²

The values of (x - 2)² and (y + 3)² are MINIMIZED when x = 2 and y = -3
So, let's take −3(x - 2)² − 2(y + 3)² − 9 and replace x with 2 and replace y with -3.
We get: −3(2 - 2)² − 2(-3 + 3)² − 9 = −3(0)² − 2(0)² − 9
= 0 - 0 - 9
= -9

Answer: B

Cheers,
Brent
Last edited by Brent@GMATPrepNow on Wed Feb 22, 2017 10:23 am, edited 1 time in total.
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by regor60 » Wed Feb 22, 2017 9:54 am
Brent@GMATPrepNow wrote:
−3x² + 12x − 2y² − 12y − 39 = −3(x² - 4x) − 2(y² − 6y) − 39
Check italicized for sign consistency
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by Brent@GMATPrepNow » Wed Feb 22, 2017 10:24 am
regor60 wrote:
Brent@GMATPrepNow wrote:
−3x² + 12x − 2y² − 12y − 39 = −3(x² - 4x) − 2(y² − 6y) − 39
Check italicized for sign consistency
Oops, missed that + sign
Thanks for the heads up.
I've edited my response accordingly.

Cheers,
Brent
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by Matt@VeritasPrep » Thu Mar 02, 2017 5:38 pm
-3x² + 12x - 2y² - 12y - 39 =>

-1 * (3x² - 12x + 2y² + 12y + 39) =>

-1 * (3*(x² - 4x) + 2*(y² + 6y) + 39) =>

-1 * (3*((x - 2)² - 4) + 2((y + 3)² - 9) + 39) =>

-1 * (3(x - 2)² - 12 + 2(y + 3)² - 18 + 39) =>

-1 * (3(x - 2)² + 2(y + 3)² + 9) =>

-9 - 3(x - 2)² - 2(y + 3)²

You'll maximize that if you aren't subtracting ANYTHING from -9, i.e. if x = 2 and y = -3. So the max is -9 when x = 2 and y = -3.
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