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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 ?

Image

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

Answer: [spoiler]______(D)__[/spoiler]
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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by fskilnik@GMATH » Thu Mar 28, 2019 11:34 am
fskilnik@GMATH 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 ?

Image

(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.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br