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Math Expert V
Joined: 02 Sep 2009
Posts: 56266

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Difficulty:   75% (hard)

Question Stats: 58% (01:45) correct 42% (01:45) wrong based on 255 sessions

### HideShow timer Statistics 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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Math Expert V
Joined: 02 Sep 2009
Posts: 56266

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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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Retired Moderator B
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Joined: 02 Apr 2014
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Location: United States
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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
Intern  B
Joined: 27 Jul 2016
Posts: 8

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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.
Intern  B
Joined: 17 May 2016
Posts: 28
GMAT 1: 740 Q46 V46 ### Show Tags

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,
Intern  Joined: 23 Apr 2016
Posts: 21
Location: Finland
GPA: 3.65

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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
Intern  B
Joined: 23 Jun 2016
Posts: 14

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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.

Retired Moderator V
Status: Long way to go!
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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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Joined: 15 Jan 2014
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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 Manager  G
Joined: 31 Jan 2017
Posts: 56
Location: India
GMAT 1: 680 Q49 V34 GPA: 4
WE: Project Management (Energy and Utilities)

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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 Manager  S
Joined: 18 Jul 2015
Posts: 50
GMAT 1: 530 Q43 V20 WE: Analyst (Consumer Products)

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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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Press +1 if my explanation helps. Thanks!
Intern  B
Joined: 20 Jul 2018
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(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.
Manager  G
Joined: 01 Feb 2017
Posts: 230

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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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Intern  B
Joined: 10 Jul 2018
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GMAT 1: 600 Q47 V26 GMAT 2: 630 Q48 V28 ### Show Tags

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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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   [#permalink] 13 Sep 2018, 09:32
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# M19-18

Moderators: chetan2u, Bunuel  