document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1079041>Question 1079041</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this <BR>\n" );
document.write( "Consider the function f(x) = x3 + 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 #693420 by </B><A HREF=/tutors/aboutme.mpl?userid=Boreal><B>Boreal(15235)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Boreal><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=Boreal><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=693420\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Derivative of the function is 3x^2+p<BR>\n" );
document.write( "Set that equal to 0 and x^2=-p/3 at the stationary point.<BR>\n" );
document.write( "F(-2)=-8-2p+q=3<BR>\n" );
document.write( "q-2p=11<BR>\n" );
document.write( "x^2=-p/3,  4=-p/3 and p=-12; <BR>\n" );
document.write( "Therefore q=-13<BR>\n" );
document.write( "x^3-12x-13<BR>\n" );
document.write( "3x^2-12=0<BR>\n" );
document.write( "x=+/-2<BR>\n" );
document.write( "when x=2, f(2)=8-24-13=-29, so (2,-29) is a point.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-10%2C10%2C-35%2C25%2Cx%5E3-12x-13%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-10%2C10%2C-35%2C25%2Cx%5E3-12x-13%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The second derivative is 6x.  <BR>\n" );
document.write( "When x<0, the second derivative is negative and there is a maximum.<BR>\n" );
document.write( "When x>0, it is positive and there is a minimum.<BR>\n" );
document.write( "the inflection point occurs at (0, -13), for there the second derivative changes sign.\r<BR>\n" );
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