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16 Sep 2014, 01:06
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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. \(a + 5\)
A. \(a-1\)
B. \(a\)
C. \(a + 2\)
D. \(a + 3\)
E. \(a + 5\)
16 Sep 2014, 01:06
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Official Solution:
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
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
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
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
