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

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swati007
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If b=a+4, then for which of the folloing [#permalink] New post   05 Sep 2013, 12:17
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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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ramannanda9
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Re: If b=a+4, then for which of the folloing [#permalink] New post   07 Sep 2013, 06:33
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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] New post   26 Dec 2014, 07:40
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If \(b = a + 4\), 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. \(a + 5\)


Given \(b = a + 4\), we can rewrite the expression \((x - a)^2 + (x - b)^2\) as \((x - a)^2 + (x - a - 4)^2\). To find the smallest value, substitute each answer choice for \(x\).

The smallest expression value is attained at \(x = a + 2\): \((x - a)^2 + (x - a - 4)^2\) becomes \((a + 2 - a)^2 + (a + 2 - a - 4)^2 = 8\).


Answer: C
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Re: If b=a+4, then for which of the folloing [#permalink] New post   05 Sep 2013, 12:39
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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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Re: If b=a+4, then for which of the folloing [#permalink] New post   11 Feb 2023, 10:10
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For a quadratic equation y = ax^2 + bx + c :

If a > 0 [Upward opening parabola]
y is min at (-b/2a)

If a < 0 [Downward opening parabola]
y is max at (-b/2a)

In this case upon getting the final equation after simplifying the squares and subtituting a = b+4,
We get -b/2a to be a+2
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Re: If b=a+4, then for which of the folloing [#permalink] New post   31 May 2023, 12:37
One quick way to solve this problem is using the number line method.

Let a=2, b= a+4 = 6

We have to make (x−a)^2+(x−b)^2 minimum. Or in other words, the sum of the squared distance of x from a and the squared distance of x from b has to be smallest.

If you draw this on the number line, you'll quickly get that this value is smallest when x is 4, i.e, x= a+2.
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Re: If b=a+4, then for which of the folloing [#permalink] New post   28 Aug 2023, 14:50
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Re: If b=a+4, then for which of the folloing [#permalink]
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