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The graphs of f(x) = x^3-x and g(x) = mx+n are represented in the fig

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New post 28 Mar 2019, 08:00
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Question Stats:

73% (02:23) correct 27% (02:27) wrong based on 22 sessions

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GMATH practice exercise (Quant Class 20)

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The graphs of \(f(x) = x^3-x\) and \(g(x) = mx+n\) are represented in the figure given. If \(m\) and \(n\) are constants, what is the value of \(mn\) ?

(A) -4
(B) -2
(C) 2
(D) 4
(E) 6

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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
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New post 28 Mar 2019, 12:31
fskilnik wrote:
GMATH practice exercise (Quant Class 20)

Image

The graphs of \(f(x) = x^3-x\) and \(g(x) = mx+n\) are represented in the figure given. If \(m\) and \(n\) are constants, what is the value of \(mn\) ?

(A) -4
(B) -2
(C) 2
(D) 4
(E) 6


Image


\(? = m \cdot n\)

\(f\left( x \right) = {x^3} - x = x\left( {{x^2} - 1} \right) = x\left( {x + 1} \right)\left( {x - 1} \right)\)

\(f\left( 2 \right) = 6\,\,\,\,\,\, \Rightarrow \,\,\,\,\,A = \left( {2,6} \right)\)

\(f\left( x \right) = 0\,\,\,\, \Rightarrow \,\,\,\,x = - 1,0,\,{\rm{or}}\,\,1\,\,\,\,\,\, \Rightarrow \,\,\,\,\,B = \left( { - 1,0} \right)\)

\(g\left( x \right) = mx + n\)

\({\rm{line}}\,\,g\,\,:\,\,\,\left\{ \matrix{
\,m = {\rm{slope}} = {{6 - 0} \over {2 - \left( { - 1} \right)}} = 2 \hfill \cr
\,B\, \in \,{\rm{graph}}\left( g \right)\,\,\,\, \Rightarrow \,\,\,\,0 = m \cdot \left( { - 1} \right) + n\,\,\,\,\, \Rightarrow \,\,\,\,\,n = 2 \hfill \cr} \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 2 \cdot 2\)


The correct answer is (D).


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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The graphs of f(x) = x^3-x and g(x) = mx+n are represented in the fig  [#permalink]

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New post 01 Apr 2019, 12:50
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Another way to look at the question:

\(g(x)\) is an upward-sloping line, whereas the intercept is positive. Hence, we can derive that \(m\) and \(n\) must be positive.

Keep in mind this information. Now let's consider the point \((2,0)\), where the functions meet at.

\(x^3 - x = mx + n\)
\(8 - 2 = 2m + n\)
\(6 = 2m + n\)

So, the equation holds for \(m = 2\) and \(n = 2\). \(2 + 2 = 4\). Pick D.
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The graphs of f(x) = x^3-x and g(x) = mx+n are represented in the fig   [#permalink] 01 Apr 2019, 12:50
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