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# If b=a+4, then for which of the folloing

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Manager
Joined: 14 Jun 2011
Posts: 61
If b=a+4, then for which of the folloing  [#permalink]

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05 Sep 2013, 11:17
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Difficulty:

65% (hard)

Question Stats:

61% (02:19) correct 39% (02:32) wrong based on 323 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

I know that we can solve this question by plugging in the option values. I want to know the smart way to choose which option to plug in first.
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Re: If b=a+4, then for which of the folloing  [#permalink]

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07 Sep 2013, 05:33
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2
Maxima and minima

Used derivatives here

(x-a)^2+(x-b)^2

First derivative=> 2*(x-a)+2*(x-b) =0 =>x=(a+b)/2 ;substitute b=> x=a+2
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Re: If b=a+4, then for which of the folloing  [#permalink]

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05 Sep 2013, 11:39
3
lets take a= 0 ; then b=4 and the equation becomes x^2+(x-4)^2. The minimum value of this function occurs when
x^2 = (x-4)^2 as both of these functions are increasing functions.

Solving this we get x=2 it is minimum. for a=0 option c is gives the value of x as 2.

hope it helps!
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Joined: 02 Sep 2009
Posts: 64891
Re: If b=a+4, then for which of the folloing  [#permalink]

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05 Sep 2013, 11:47
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swati007 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

I know that we can solve this question by plugging in the option values. I want to know the smart way to choose which option to plug in first.

Discussed here: m19-q18-120286.html
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Math Expert
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Posts: 64891
Re: If b=a+4, then for which of the folloing  [#permalink]

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26 Dec 2014, 06:40
1
swati007 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

I know that we can solve this question by plugging in the option values. I want to know the smart way to choose which option to plug in first.

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 if 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: If b=a+4, then for which of the folloing  [#permalink]

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27 Aug 2019, 09:12
swati007 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

I know that we can solve this question by plugging in the option values. I want to know the smart way to choose which option to plug in first.

Given: b=a+4

Asked: For which of the following values of x is the expression (x−a)^2+(x−b)^2 the smallest?

(x−a)^2+(x−b)^2 = 2x^2 - 2(a+b)x + a^2 + b^2 = 2{x^2 - (a+b)x + (a+b)^2/4} - (a+b)^2/2 + a^2 + b^2 = 2(x - (a+b)/2))^2 + - (a+b)^2/2 + a^2 + b^2

For the expression to be the smallest
x = (a+b)/2 = (2a +4)/2 = a + 2

IMO C
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Re: If b=a+4, then for which of the folloing   [#permalink] 27 Aug 2019, 09:12

# If b=a+4, then for which of the folloing

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