fskilnik wrote:
GMATH practice exercise (Quant Class 13)
The curve shown above is defined by the ordered-pairs (x,y) such that y = f(x) = Ax^2+
2Bx+C, where A, B and C are given constants. If the point of tangency with the x-axis has a positive x-coordinate, which of the following must be true?
I. A and C are both positive.
II. B^2 is greater than twice the value of AC.
III. AC/B is negative.
(A) I only
(B) I and II only
(C) I and III only
(D) All of them
(E) None of them
\(y = A{x^2} + 2Bx + C\)
\(A > 0\,\,:\,\,\,{\rm{parabola}}\,\,{\rm{concave}}\,\,{\rm{upward}}\,\)
\(C > 0\,\,:\,\,\,y - {\rm{intercept}}\,\,{\rm{ > }}\,\,{\rm{0}}\,\,\,\,\,\,\,\,\left[ {f\left( 0 \right) = A \cdot {0^2} + 2B \cdot 0 + C\,\,\, \Rightarrow \,\,\,\left( {0,C} \right) \in {\rm{curve}}} \right]\)
\({\rm{tangency}}\,\,:\,\,0 = \Delta = {\left( {2B} \right)^2} - 4AC = 4\left( {{B^2} - AC} \right)\,\,\,\,\, \Rightarrow \,\,\,{B^2} = AC\)
\({\rm{I}}.\,\,A,C\,\,\mathop > \limits^? \,\,0\,\,\,\left[ {{\rm{True}}} \right]\)
\({\rm{II}}{\rm{.}}\,\,{B^2}\,\,\mathop > \limits^? \,\,2AC\,\,\,\left[ {{\rm{False}}} \right]\,\,\,:\,\,\,{B^2} = AC\,\,\mathop < \limits^{AC\, > \,0} 2AC\)
\({\rm{III}}{\rm{.}}\,\,{{AC} \over B}\,\,\mathop = \limits^{\left( * \right)} \,\,B\,\,\mathop < \limits^? \,\,0\,\,\,\left[ {{\rm{True}}} \right]\,\,\,:\,\,\,0\mathop < \limits^{{\rm{stem!}}} {x_{{\rm{vert}}}} = - {{2B} \over {2A}} = - {B \over A}\,\,\,\,\,\mathop \Rightarrow \limits^{A\, > \,0} \,\,\,\,B < 0\)
\(\left( * \right)\,\,B = 0\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \matrix{\\
\,AC = {B^2} = 0\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,A\,\,{\rm{or}}\,\,C\,\,{\rm{zero}}\,,\,\,{\rm{impossible}} \hfill \cr \\
\,y = f\left( x \right) = A{x^2} + C\,\,\,\,\, \Rightarrow \,\,\,\,y{\rm{ - axis}}\,\,{\rm{is}}\,\,{\rm{symmetry}}\,\,{\rm{axis}}\,{\rm{,}}\,\,{\rm{impossible}}\,\,\,\left( {{\rm{stem}}} \right) \hfill \cr} \right.\)
The correct answer is (C).
We follow the notations and rationale taught in the
GMATH method.
Regards,
Fabio.