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M19-18

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M19-18  [#permalink]

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New post 16 Sep 2014, 01:06
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59% (01:38) correct 41% (01:40) wrong based on 216 sessions

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If \(b=a+4\), then for which of the following values of \(x\) is the expression \((x-a)^2 + (x-b)^2\) the smallest?

A. \(a-1\)
B. \(a\)
C. \(a + 2\)
D. \(a + 3\)
E. \(a + 5\)

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Re M19-18  [#permalink]

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New post 16 Sep 2014, 01:06
Official Solution:

If \(b=a+4\), then for which of the following values of \(x\) is the expression \((x-a)^2 + (x-b)^2\) the smallest?

A. \(a-1\)
B. \(a\)
C. \(a + 2\)
D. \(a + 3\)
E. \(a + 5\)


Since \(b=a+4\) then \((x-a)^2 + (x-b)^2=(x-a)^2 + (x-a-4)^2\). Now, plug each option for \(x\) to see which gives the least value.

The least value of the expression is for \(x=a+2\): \((x-a)^2 + (x-a-4)^2=(a+2-a)^2 + (a+2-a-4)^2=8\).


Answer: C
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Re: M19-18  [#permalink]

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New post 24 Jul 2016, 18:10
Bunuel wrote:
Official Solution:

If \(b=a+4\), then for which of the following values of \(x\) is the expression \((x-a)^2 + (x-b)^2\) the smallest?

A. \(a-1\)
B. \(a\)
C. \(a + 2\)
D. \(a + 3\)
E. \(a + 5\)


Since \(b=a+4\) then \((x-a)^2 + (x-b)^2=(x-a)^2 + (x-a-4)^2\). Now, plug each option for \(x\) to see which gives the least value.

The least value of the expression is for \(x=a+2\): \((x-a)^2 + (x-a-4)^2=(a+2-a)^2 + (a+2-a-4)^2=8\).


Answer: C




hi
please throw some light how you picked the least value for x as a-2?
any pattern or logic via looking at answer choices?
thanks
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Re: M19-18  [#permalink]

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New post 27 Jul 2016, 23:07
There's a simple trick for guys who are having a problem with finding the maximum or the minimum value question(and it saves a lot of time)

differentiate the equation and then equate to zero. the values of X will be either maximun or minimum value. to determine whether its max or min put any number for x(i prefer 0). Dont worry about point of inflexion as it wont be asked in GMAT. For simple differentiation rules look up google. Its pretty simple.
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Re: M19-18  [#permalink]

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New post 02 Aug 2016, 01:46
ABHISHEK8998 wrote:
There's a simple trick for guys who are having a problem with finding the maximum or the minimum value question(and it saves a lot of time)

differentiate the equation and then equate to zero. the values of X will be either maximun or minimum value. to determine whether its max or min put any number for x(i prefer 0). Dont worry about point of inflexion as it wont be asked in GMAT. For simple differentiation rules look up google. Its pretty simple.



Abshishek, thank you for the tip, could you please detail your solution ?
Many thanks in advance,

Best regards,
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Re: M19-18  [#permalink]

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New post 14 Oct 2016, 10:28
nickimonckom
if you diffrentiate (x-a)^2 + (x-b)^2, you get 2(x-a)+2(x-b)
Now if you equate this with 0
2(x-a)+2(x-b) = 0 gives you x = (a+b)/2, if you plug b = a+4, you get x = a+2
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Re: M19-18  [#permalink]

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New post 17 Dec 2016, 05:37
nickimonckom wrote:
ABHISHEK8998 wrote:
There's a simple trick for guys who are having a problem with finding the maximum or the minimum value question(and it saves a lot of time)

differentiate the equation and then equate to zero. the values of X will be either maximun or minimum value. to determine whether its max or min put any number for x(i prefer 0). Dont worry about point of inflexion as it wont be asked in GMAT. For simple differentiation rules look up google. Its pretty simple.



Abshishek, thank you for the tip, could you please detail your solution ?
Many thanks in advance,

Best regards,


Your strategy helps.

Could you please specify how to determine whether it is min value or max value in more detail.

Thanks is advance.
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Re: M19-18  [#permalink]

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New post 17 Dec 2016, 11:08
Bunuel wrote:
If \(b=a+4\), then for which of the following values of \(x\) is the expression \((x-a)^2 + (x-b)^2\) the smallest?

A. \(a-1\)
B. \(a\)
C. \(a + 2\)
D. \(a + 3\)
E. \(a + 5\)


We could use AM-GM inequality to solve this easily.

\((x-a)^2+(x-b)^2=(x-a)^2+(b-x)^2 \geq \frac{1}{2}[(x-a)+(b-x)]^2=\frac{1}{2}(b-a)^2=\frac{4^2}{2}=8\)

\(min((x-a)^2+(x-b)^2)=8 \iff x-a=b-x \iff x=\frac{a+b}{2}=a+2\)

The answer is C
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Re: M19-18  [#permalink]

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New post 17 Feb 2017, 18:49
Another way of doing this question :

We can use this property : \((A-B)^2 = A^2+B^2-2AB\)

\(A = (x−a)^2\) ; \(B = (x−b)^2\)

\((x-a-x+b)^2 = (x−a)^2 + (x−b)^2 -2(x-a)(x-b)\)

==> \((b-a)^2 +2(x-a)(x-b) = (x−a)^2 + (x−b)^2\)

Given b-a = 4 ;

\(16 +2(x-a)(x-b) = (x−a)^2 + (x−b)^2\)
\(16 +2(x-a)(x-a-4) = (x−a)^2 + (x−b)^2\) . Now we need to minimize this expression .So wil try to minimize value for \(2(x-a)(x-a-4)\) to 0 or negative .

\(x = a\) will make this value 0 and this will give 16 ;
\(x=a+2\) will make above expression -ve and will give the value less than 16 hence our answer.

Also, notice that next value of \(x=a+3\) will make \(2(x-a)(x-a-4)\) +ve and hence will make overall value bigger.


Please +1 kudos if this post helps :)
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Re: M19-18  [#permalink]

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New post 27 Feb 2017, 09:41
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Answer: C

Expression= (x−a)^2+(x−b)^2 = 2x^2 - 2ax -2bx +a^2 + b^2

For extreme values of expression, d/dx(expression) =0

Differentiating,
d/dx(expression) = 4x -2a -2b = 0

or, 4x= 2a+2b = 4a+8 [putting values of b=a+4]
Therefore, x= a+2

It can be ascertained that this extreme value is minimum as d^2/dx^2(expression) = 4 (positive)

Note: Differentiation is not tested on the GMAT, but following little information helps

1. d/dx(constant) = 0
2. d/dx(x^n) = n * x^(n-1)
3. d/dx(c*x) = c ; where c = constant
4. Expression has extreme values at d/dx(expression)=0
; d2/dx^2(expression) = positive signifies minimum value
; d2/dx^2(expression) = negative signifies maximum value
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M19-18  [#permalink]

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New post 16 Sep 2017, 00:29
We can also pick numbers to solve this problem. Below is the approach I used:
Let's take \(a = 1\), hence \(b\) will be 5 (\(b = a + 4\)). Now substitute the value of \(a\) in each of the answer options to get the value of \(x\).

Equation: \((x-a)^2 + (x-b)^2\)

A) \(a-1\), \(x=0\), substitute in the above equation; result will be \(26\)
B) \(a\), \(x=1\), \(16\)
C) \(a+2\), \(x=3\), 8 - Smallest Value (Correct Ans.)
D) \(a+3\), \(x=4\), \(10\)
E) \(a+5\), \(x=6\), \(26\)
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Re: M19-18  [#permalink]

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New post 20 Jul 2018, 14:50
(x-a)^2 is a standard parabola moved to the right by a.
(x-(a+4))^2 is a standard parabola moved to the right by a + 4.

the total formula is the sum of these two parabolas, which is smallest in the middle between the two minima a and a + 4.
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Re: M19-18  [#permalink]

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New post 28 Jul 2018, 15:03
Plug in:
Assume a=2
APS, b=6
Expression: (x-2)^2+(x-6)^2
Expanding: 2(x^2+20-8x)

A) a-1=1, expression value:2*13
B) a=2, expression value:2*8
C) a+2=4, expression value:2*4
D) a+3=5, expression value:2*5
E) a+5=7, expression value:2*13

Min value: Choice C

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Re: M19-18  [#permalink]

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New post 31 Jul 2018, 17:42
If you draw a graph, you'll find out that the midpoint from a and a+4 will be where the x is lowest...
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Re M19-18  [#permalink]

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New post 13 Sep 2018, 09:32
I think this is a high-quality question and I agree with explanation. Assume a = 0
(x)2+(x−4)2

-1 --- 10
0 --- 16
2 --- 8
3 --- 10
5 ---- 26

So the answer is option C
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Re M19-18 &nbs [#permalink] 13 Sep 2018, 09:32
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