document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=285986>Question 285986</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do I find the stationary points in this equation:\r<BR>\n" );
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document.write( "y=(3x-1)(x-2)^4 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #207391 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=207391\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=”Garamond” size=”+2”>\r<BR>\n" );
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document.write( "There are 2 ways to go about this.\r<BR>\n" );
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document.write( "The bull-in-a-china-shop method is to multiply out the five factors and then differentiate the resulting 5th degree polynomial function.  Not too difficult really, just a bit of drudgery.\r<BR>\n" );
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document.write( "The other way is to apply the Chain Rule and the Product Rule to find the first derivative of the function. Certainly the more elegant way to do it.\r<BR>\n" );
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document.write( "Either way, once you have the first derivative, set it equal to zero.  There will be a stationary point at every zero of the first derivative function.\r<BR>\n" );
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document.write( "Once you have the first derivative, you will note that it is a rather ugly 4th degree polynomial.  But if you look at a graph of the original function (use a graphing application or your graphing calculator, or simply realize that the 5th degree polynomial has a zero at 2 with a multiplicity of 4) you will see that there are only two stationary points, one at 2 and one between 0 and 1, just about 2/3.\r<BR>\n" );
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document.write( "If you take the ugly mess of a 4th degree polynomial function that you get for a first derivative, you will find that you will be successful performing synthetic division three successive times with a divisor of 2.  That will leave you with a simple linear equation the solution of which is the remaining stationary point.\r<BR>\n" );
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document.write( "Good luck\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large\ e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
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