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# M19-18

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Math Expert
Joined: 02 Sep 2009
Posts: 51071

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16 Sep 2014, 00:06
00:00

Difficulty:

75% (hard)

Question Stats:

59% (01:39) correct 41% (01:40) wrong based on 225 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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Joined: 02 Sep 2009
Posts: 51071

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16 Sep 2014, 00:06
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 $$x=a+2$$: $$(x-a)^2 + (x-a-4)^2=(a+2-a)^2 + (a+2-a-4)^2=8$$.

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24 Jul 2016, 17:10
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 $$x=a+2$$: $$(x-a)^2 + (x-a-4)^2=(a+2-a)^2 + (a+2-a-4)^2=8$$.

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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27 Jul 2016, 22:07
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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Joined: 17 May 2016
Posts: 29
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02 Aug 2016, 00:46
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 ?

Best regards,
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14 Oct 2016, 09:28
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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Joined: 23 Jun 2016
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17 Dec 2016, 04:37
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 ?

Best regards,

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

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17 Dec 2016, 10:08
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.

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

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

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17 Feb 2017, 17:49
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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27 Feb 2017, 08:41
2
2

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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Kindly press "+1 Kudos" if the post helped

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15 Sep 2017, 23:29
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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20 Jul 2018, 13:50
(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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28 Jul 2018, 14:03
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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Joined: 10 Jul 2018
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31 Jul 2018, 16:42
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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13 Sep 2018, 08:32
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
Re M19-18 &nbs [#permalink] 13 Sep 2018, 08:32
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# M19-18

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