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If f(x)=x^2/b^2 + 2x + 4, then for each non-zero value of b

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Revenge2013
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If f(x)=x^2/b^2 + 2x + 4, then for each non-zero value of b [#permalink] New post   01 Sep 2013, 10:35
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If f(x)=x^2/b^2 + 2x + 4, then for each non-zero value of b the function of f(x) is the least when x equals

A. \(-1 -\sqrt{(1-4b)}\)
B. -2
C. 0
D. -b^2
E. b-4
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D
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Bunuel
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Re: If f(x)=x^2/b^2 + 2x + 4, then for each non-zero value of b [#permalink] New post   01 Sep 2013, 11:56
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Revenge2013 wrote:
If f(x)=x^2/b^2 + 2x + 4, then for each non-zero value of b the function of f(x) is the least when x equals

A. \(-1 -\sqrt{(1-4b)}\)
B. -2
C. 0
D. -b^2
E. b-4


Complete the square.

\(f(x)=\frac{x^2}{b^2} + 2x + 4\) --> add b^2-b^2 --> \(f(x)=\frac{x^2}{b^2} + 2x + b^2 +4 -b^2\) --> \(f(x)=(\frac{x}{b})^2+ 2x + b^2 +4 -b^2\) --> \(f(x)=(\frac{x}{b}+b)^2+(4 -b^2)=(non-negative)+(constant)\).

The least value of f(x) when \((\frac{x}{b}+b)^2=0\), so when \(\frac{x}{b}+b=0\) or when \(x=-b^2\).

Answer: D.
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Re: If f(x)=x^2/b^2 + 2x + 4, then for each non-zero value of b [#permalink] New post   09 May 2015, 16:36
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I suppose you could solve by looking at the answer choices. If the answer is going to be correct for absolutely every value of b, it certainly must be correct when b=1. So we can let b=1 and just plug each answer choice into

x^2 + 2x + 4

to see which gives us the least value of this function. Notice answer A is completely undefined (we get a negative under the root), so cannot possibly be right. If we plug in -2 or 0, the value of the function is 4 in both cases. If we plug in answer D, which is -b^2 = -1, the function's value is 3, which is the smallest value so far. Finally if we plug in answer E, which would be equal to b - 4 = 1 - 4 = -3, the function's value is 6. So the correct answer is -b^2.

But if the GMAT were to ask a question like this, there would always be a way to correctly solve it 'from start to finish', without looking at the answer choices. And I don't see a way to do that here using normal GMAT math. I've never once needed to 'complete the square' in any official GMAT question I've ever solved, nor have I ever needed the quadratic formula, or needed to know how to find the minimum value of a general parabola. I've solved probably close to 10,000 official questions by now, so those techniques just aren't ever required on the test. This question is one that would normally be answered using calculus anyway (and is quite easy if you know calculus), so it's a question I'd bet GMAC would consider unfair, because people with certain educational backgrounds would have a big advantage answering it, and the GMAT is not supposed to be a test of whether you've ever taken a calculus class. It's a test of how well you can reason using only the most elementary facts of mathematics.

So unless I'm not seeing a solution here that uses normal GMAT math, there's really no reason to worry about this question.
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Revenge2013
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Re: If f(x)=x^2/b^2 + 2x + 4, then for each non-zero value of b [#permalink] New post   01 Sep 2013, 14:08
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Bunuel wrote:
Revenge2013 wrote:
If f(x)=x^2/b^2 + 2x + 4, then for each non-zero value of b the function of f(x) is the least when x equals

A. \(-1 -\sqrt{(1-4b)}\)
B. -2
C. 0
D. -b^2
E. b-4


Complete the square.

\(f(x)=\frac{x^2}{b^2} + 2x + 4\) --> add b^2-b^2 --> \(f(x)=\frac{x^2}{b^2} + 2x + b^2 +4 -b^2\) --> \(f(x)=(\frac{x}{b})^2+ 2x + b^2 +4 -b^2\) --> \(f(x)=(\frac{x}{b}+b)^2+(4 -b^2)=(non-negative)+(constant)\).

The least value of f(x) when \((\frac{x}{b}+b)^2=0\), so when \(\frac{x}{b}+b=0\) or when \(x=-b^2\).

Answer: D.


Bunuel,

Thanks for the quick reply as always!

I still fail to see the (incomplete) square that should have triggered something in me to come up with b^2-b^2. Was the trigger the x^2/b^2?
The rest is clear.

Many thanks!
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Re: If f(x)=x^2/b^2 + 2x + 4, then for each non-zero value of b [#permalink] New post   02 Sep 2013, 06:03
Hi Bunuel,

I am still not clear on how you arrived on the following:

f(x)=(\frac{x}{b}+b)^2+(4 -b^2)=(non-negative)+(constant).

The least value of f(x) when (\frac{x}{b}+b)^2=0, so when \frac{x}{b}+b=0 or when x=-b^2.

can you please elaborate?
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Re: If f(x)=x^2/b^2 + 2x + 4, then for each non-zero value of b [#permalink] New post   02 Sep 2013, 09:15
Bunuel wrote:
Revenge2013 wrote:
If f(x)=x^2/b^2 + 2x + 4, then for each non-zero value of b the function of f(x) is the least when x equals

A. \(-1 -\sqrt{(1-4b)}\)
B. -2
C. 0
D. -b^2
E. b-4


Complete the square.

\(f(x)=\frac{x^2}{b^2} + 2x + 4\) --> add b^2-b^2 --> \(f(x)=\frac{x^2}{b^2} + 2x + b^2 +4 -b^2\) --> \(f(x)=(\frac{x}{b})^2+ 2x + b^2 +4 -b^2\) --> \(f(x)=(\frac{x}{b}+b)^2+(4 -b^2)=(non-negative)+(constant)\).

The least value of f(x) when \((\frac{x}{b}+b)^2=0\), so when \(\frac{x}{b}+b=0\) or when \(x=-b^2\).

Answer: D.



Bunuel,

Can we say the quadratic equation ax^2 + bx + c reaches minimum for -b/2a

so the OA is D
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Re: If f(x)=x^2/b^2 + 2x + 4, then for each non-zero value of b [#permalink] New post   03 Sep 2013, 00:43
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In this question, we can see that the coefficient of x^2 is always positive, therefore the equation is a parabola facing upwards in a coordinate plane. In a parabola facing upwards there is only minima (no maxima) which is equal to (-coeff of x/2coeff of x^2), in this case -b^2. Hence the answer, D.

Some basic knowledge about coordinate geometry makes such questions cake walk.
The GMAT Club Math Book deals with such basics appropriately.

Hope it helps.
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Re: If f(x)=x^2/b^2 + 2x + 4, then for each non-zero value of b [#permalink] New post   07 Sep 2013, 11:45
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Derivatives do the trick here as well

f(x)=x^2/b^2+2x+4
=> 2x/b^2+2=0 => x=-b^2 which gives a minimum value of -b^2+4
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Re: If f(x)=x^2/b^2 + 2x + 4, then for each non-zero value of b [#permalink] New post   09 May 2015, 14:59
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Is there a way to solve this problem without adding b^2 - b^2?
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Re: If f(x)=x^2/b^2 + 2x + 4, then for each non-zero value of b [#permalink] New post   22 Apr 2018, 05:53
Revenge2013 wrote:
If f(x)=x^2/b^2 + 2x + 4, then for each non-zero value of b the function of f(x) is the least when x equals

A. \(-1 -\sqrt{(1-4b)}\)
B. -2
C. 0
D. -b^2
E. b-4


This question can also be solved by considering the derivative of f(x).
The theory states that the function can achieve its maximum or minimum value at x when d(f(x)/dx= 0 (basically slope of the curve is 0, parallel to x axis)
Now to find out whether the function gets max or min we need to find out double derivative of f(x)
if the double derivative of f(x) at x which is calculated above from single derivative is <0 then the function attains maximum value
otherwise the function attains minimum value.
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Re: If f(x)=x^2/b^2 + 2x + 4, then for each non-zero value of b [#permalink] New post   15 Mar 2024, 06:27
Since a is positive in the equation, parabola is facing up. So value of x at the turning point is x = -b/2a

-2 /1 x b^2 /2 = -b^2

Adewale Fasipe GMAT, GRE quant instructor from Lagos, Nigeria.
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Re: If f(x)=x^2/b^2 + 2x + 4, then for each non-zero value of b [#permalink]
15 Mar 2024, 06:27
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