document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1079044>Question 1079044</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this<BR>\n" );
document.write( "Consider the function f(x) = x^3 + px + q.<BR>\n" );
document.write( "(a) Determine the values of p and q if f(x) has a stationary point at (&#8722;2,3).<BR>\n" );
document.write( "(b) Show, using calculus, that there is a second stationary point at (2,&#8722;29), and classify both<BR>\n" );
document.write( "stationary points.<BR>\n" );
document.write( "(c) Determine f&#8242;&#8242;(x) and hence show that there is a non-stationary point of inflection and determine its coordinates.<BR>\n" );
document.write( "(d) For what values of k would the equation f(x) = k have 3 distinct solutions. Give reasons for your answer.<BR>\n" );
document.write( "(Hint: Sketch the graph of y = f(x).) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #693408 by </B><A HREF=/tutors/aboutme.mpl?userid=rothauserc><B>rothauserc(4718)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rothauserc><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=rothauserc><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=693408\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A stationary point is where the first derivative is defined and equal to zero<BR>\n" );
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document.write( "f'(x) = 3x^2 + p<BR>\n" );
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document.write( "we know x = -2 at the stationary point (-2,3)<BR>\n" );
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document.write( "1)  3(-2)^2 + p = 0<BR>\n" );
document.write( "p = -12<BR>\n" );
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document.write( "we know f(-2) = 3 since (-2,3) is a point on the graph of f(x)<BR>\n" );
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document.write( "2) 3 = (-2)^3 -12(-2) + q<BR>\n" );
document.write( "3 = -8 +24 + q<BR>\n" );
document.write( "q = -13<BR>\n" );
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document.write( "f(x) = x^3 -12x -13<BR>\n" );
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document.write( "a.  p = -12, q = -13<BR>\n" );
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document.write( "f'(x) = 3x^2 -12<BR>\n" );
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document.write( "3x^2 -12 = 0<BR>\n" );
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document.write( "3x^2 = 12<BR>\n" );
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document.write( "x^2 = 4<BR>\n" );
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document.write( "x = 2 and x = -2<BR>\n" );
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document.write( "b. for x = 2<BR>\n" );
document.write( "f(2) = 2^3 -12(2) -13 = -29<BR>\n" );
document.write( "stationary point at (2,-29)<BR>\n" );
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document.write( "f'(x) = 3x^2 -12<BR>\n" );
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document.write( "An inflection point is a point on a curve where the curve changes sign.<BR>\n" );
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document.write( "we take the second derivative, which is<BR>\n" );
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document.write( "f''(x) = 6x<BR>\n" );
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document.write( "our candidate x value is x = 0<BR>\n" );
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document.write( "Note that if x is negative then 6x is negative and if x is positive then 6x is positive, therefore the point with x = 0 is an inflection point<BR>\n" );
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document.write( "f(0) =  0^3 -12(0) -13 = -13<BR>\n" );
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document.write( "c.  the point of inflection is (0,-13)<BR>\n" );
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document.write( "the graph of f(x) = x^3 -12x -13 looks like<BR>\n" );
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document.write( "we want values of k, such that f(x) crosses the x axis 3 times<BR>\n" );
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document.write( "d.  looking at the graph, this is where f(x) = 0, that is, k = 0<BR>\n" );
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