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Taking derivatives on GMAT questions

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Taking derivatives on GMAT questions [#permalink] New post 14 Nov 2011, 10:55
If A and B are integers and B = A + 4, which of the following represents integer X for which the expression (X - A)^2 + (X - B)^2 is the smallest?

A) A-1
B) A
C) A+2
D) A+3
E) B+1

OA C

Here was an answer explanation that used calculus and worked...does anyone know of any other examples where calculus could be used or know in general the type of question where calc. might come in handy?

18 seconds method, take the derivatives of both polynomials of order two to get 2 ( 2x - A - B) = 0, now substitute for B as given in the question and get the value of X which is X = A + 2
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Re: Taking derivatives on GMAT questions [#permalink] New post 14 Nov 2011, 12:48
Expert's post
I am still learning how to use the forum features so read carefully as I was unable to do the math formulas

I dont know if this is a GMAT question.. but here is what is how I would approach this by logic.

1. Note that in this expression you have a spread of 4 (the difference between A and B) to be distributed in 2 parts.
2. The distribution should be such that sum of squares is minimized. The minimum would be 2 (square) + 2 (square) = 8

Hence, X = A+2

-------------------------------------------------------------------------------------------------------------------------------------------------
Alternate approach

Let X = A+K, where K is a constant

Expression: (A+K-A)^2 + (A+K-A-4)^2

Simplifying: k^2 + (k-4)^2

This is minimum for k = 2


I hope this helps.

-Rajat


Alternate question

What would be the value of X for (X - A)^2 - (X - B)^2 to be smallest? What is the smallest value.
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Re: Taking derivatives on GMAT questions [#permalink] New post 14 Nov 2011, 20:20
Expert's post
stringworm wrote:
If A and B are integers and B = A + 4, which of the following represents integer X for which the expression (X - A)^2 + (X - B)^2 is the smallest?

A) A-1
B) A
C) A+2
D) A+3
E) B+1

OA C

Here was an answer explanation that used calculus and worked...does anyone know of any other examples where calculus could be used or know in general the type of question where calc. might come in handy?

18 seconds method, take the derivatives of both polynomials of order two to get 2 ( 2x - A - B) = 0, now substitute for B as given in the question and get the value of X which is X = A + 2


Derivatives can be helpful in finding the maximum/minimum value of a polynomial if you don't want to think much. In GMAT, such questions would be easily solvable if you just use some logic. You don't need to do any long calculations. I would suggest you to not waste your time revisiting calculus.

The following would be my reasoning in this question:
We need to minimize the sum of two squares. Each square term should be minimized to minimize the sum. A square has minimum value of 0. I cannot make both the terms 0 since x cannot be A and B at the same time. A and B differ by 4 so I should take x right in the middle i.e. (A + 2) so that both the terms, (x-A) and (x-B) and as small as possible simultaneously. Then their squares will be as small as possible and hence the sum will be minimized too.
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Re: Taking derivatives on GMAT questions   [#permalink] 14 Nov 2011, 20:20
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