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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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28 Mar 2019, 08:00
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Difficulty:

35% (medium)

Question Stats:

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

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GMATH practice exercise (Quant Class 20)

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)
Our high-level "quant" preparation starts here: https://gmath.net
GMATH Teacher
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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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28 Mar 2019, 12:31
fskilnik wrote:
GMATH practice exercise (Quant Class 20)

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

$$? = m \cdot n$$

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

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

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

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

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

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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01 Apr 2019, 12:50
1
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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