document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=175746>Question 175746</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0618250204/algebrahomeworkh>Algebra 2</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> How do I graph this polynomial function? f(x)=x^4 - 2x^3 - 3x^2 + 5x + 2.<BR>\n" );
document.write( "Thank you very much. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #130891 by </B><A HREF=/tutors/aboutme.mpl?userid=nabla><B>nabla(475)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=nabla><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=130891\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1) Factor to f(x)=(-2 + x) (-1 - 3 x + x^3)<BR>\n" );
document.write( "2) The first evident, real root will be x=2.<BR>\n" );
document.write( "3) It should be noted that this will \"eventually\" resemble a parabola. Moreover, our interest will be around x=2.<BR>\n" );
document.write( "4) Techniques of calculus will give us concavity, max, min, inflection.\r<BR>\n" );
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document.write( "f'(x)=4x^3-6x^2-6x+5=0 gives criticals of\r<BR>\n" );
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document.write( "x -> -1.05274<BR>\n" );
document.write( "x -> 0.611731<BR>\n" );
document.write( "x -> 1.94101\r<BR>\n" );
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document.write( "If you test outside the intervals you will note changes in direction of the slope this is indicative of extrema.<BR>\n" );
document.write( "Thus: x=-1.05274 is local minimum<BR>\n" );
document.write( "x=0.611731 is local maximum<BR>\n" );
document.write( "x=1.94101 is local minimum\r<BR>\n" );
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document.write( "Concavity [not entirely necessary]:<BR>\n" );
document.write( "f''(x)=12x^2-12x-6=0<BR>\n" );
document.write( "gives<BR>\n" );
document.write( "x -> -0.366025<BR>\n" );
document.write( "x -> 1.36603\r<BR>\n" );
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document.write( "Testing intervals again you will find CU on (-inf, -0.366025), CD on (-0.366025, 1.36603), then CU on (1.36603, inf).\r<BR>\n" );
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document.write( "This is certainly enough information to produce the following graph:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+300%2C+200%2C+-3%2C+4%2C+-5%2C+10%2C+x%5E4-2x%5E3-3x%5E2%2B5x%2B2%29 ALIGN=MIDDLE  ALT=\"graph%28+300%2C+200%2C+-3%2C+4%2C+-5%2C+10%2C+x%5E4-2x%5E3-3x%5E2%2B5x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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