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If f(x) = 4x^2 + px, what is the minimum value of f(x)?

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If f(x) = 4x^2 + px, what is the minimum value of f(x)?  [#permalink]

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New post 04 Jul 2018, 01:52
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[GMAT math practice question]

If \(f(x) = 4x^2 + px\), what is the minimum value of \(f(x)\)?

\(1) f(1) = -4\)
\(2) f(2) = 0\)

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Re: If f(x) = 4x^2 + px, what is the minimum value of f(x)?  [#permalink]

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New post 04 Jul 2018, 02:41
MathRevolution wrote:
[GMAT math practice question]

If \(f(x) = 4x^2 + px\), what is the minimum value of \(f(x)\)?

\(1) f(1) = -4\)
\(2) f(2) = 0\)



Minimum value of \(4x^2+px\) is at 8x+p=0 or when \(x=\frac{-p}{8}\)
So min value can be found if value of p is known...

1) f(1)=-4
f(x)=4x^2+px so f(1)=4*1^2+p*1=4+p=-4........ p=-8
So min value be at x=-8/-8=1
f(x)=4x^2-8x=4*(1)^2-8*1=-4
Sufficient

2)f(2)=0
f(x)=4x^2+px so f(2)=4*2^2+p*2=16+2p=0....... p=16/(-2)=-8
So min value be at x=-8/-8=1
f(x)=4x^2-8x=4*(1)^2-8*1=-4
Sufficient

D
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Re: If f(x) = 4x^2 + px, what is the minimum value of f(x)?  [#permalink]

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New post 04 Jul 2018, 02:52
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MathRevolution wrote:
[GMAT math practice question]

If \(f(x) = 4x^2 + px\), what is the minimum value of \(f(x)\)?

\(1) f(1) = -4\)
\(2) f(2) = 0\)


To determine the minimum value of the graph, we need to know the value of p.
Question stem, rephrased:
What is the value of p?

Statement 1:
Substituting \(x=1\) and \(f(x)=-4\) into \(f(x) = 4x^2 + px\), we get:
\(-4 = 4(1^2) + p(1)\)
Since we can solve for \(p\), SUFFICIENT.

Statement 2:
Substituting \(x=2\) and \(f(x)=0\) into \(f(x) = 4x^2 + px\), we get:
\(0 = 4(2^2) + p(2)\)
Since we can solve for \(p\), SUFFICIENT.


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Re: If f(x) = 4x^2 + px, what is the minimum value of f(x)?  [#permalink]

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New post 06 Jul 2018, 01:41
=>
Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 1 variable (p) and 0 equations, D is most likely to be the answer. So, we should consider each of the conditions on their own first.

Condition 1)
Since \(f(1) = -4,\) we have \(4 + p = -4\) and p = \(-8.\)
\(f(x) = 4x^2 -8x = 4( x^2 - 2x + 1 - 1 ) = 4(x-1)^2 – 4.\)
The minimum of \(f(x)\) is \(-4\).
Condition 1) is sufficient.

Condition 2)
Since \(f(2) = 0\), we have \(16 + 2p = 0\) and \(p = -8.\)
\(f(x) = 4x^2 -8x = 4( x^2 - 2x + 1 - 1 ) = 4(x-1)^2 – 4.\)
The minimum of \(f(x)\) is \(-4.\)
Condition 2) is sufficient.

Therefore, D is the answer.

Answer: D

Note: Tip 1) of the VA method states that D is most likely to be the answer if condition 1) gives the same information as condition 2).


If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
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Re: If f(x) = 4x^2 + px, what is the minimum value of f(x)? &nbs [#permalink] 06 Jul 2018, 01:41
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