document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1180886>Question 1180886</A>: <I><SPAN STYLE=\"word-break: break-all;\"> At what values of x on the curve y=5+40x^3-3x^5 does the tangent line have the largest slope?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #810702 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=810702\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "At what values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the curve \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5%2B40x%5E3-3x%5E5 ALIGN=MIDDLE  ALT=\"y=5%2B40x%5E3-3x%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" > does the tangent line have the largest slope?\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28dy%2Fdx%29%285%2B40x%5E3-3x%5E5%29=3%2A40x%5E2-5%2A3x%5E4 ALIGN=MIDDLE  ALT=\"%28dy%2Fdx%29%285%2B40x%5E3-3x%5E5%29=3%2A40x%5E2-5%2A3x%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28dy%2Fdx%29%285%2B40x%5E3-3x%5E5%29=120x%5E2-15x%5E4 ALIGN=MIDDLE  ALT=\"%28dy%2Fdx%29%285%2B40x%5E3-3x%5E5%29=120x%5E2-15x%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28d%5E2y%2Fdx%5E2%29%285%2B40x%5E3-3x%5E5%29=240x-60x%5E3 ALIGN=MIDDLE  ALT=\"%28d%5E2y%2Fdx%5E2%29%285%2B40x%5E3-3x%5E5%29=240x-60x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "To find where the slope of the tangent line is the largest, we need to find the inflection points which occur when the second derivative equals zero:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=240x-60x%5E3=0 ALIGN=MIDDLE  ALT=\"240x-60x%5E3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60x%284-x%5E2%29=0 ALIGN=MIDDLE  ALT=\"60x%284-x%5E2%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "solutions:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60x=0 ALIGN=MIDDLE  ALT=\"60x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4-x%5E2=0 ALIGN=MIDDLE  ALT=\"4-x%5E2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282-x%29%282%2Bx%29=0 ALIGN=MIDDLE  ALT=\"%282-x%29%282%2Bx%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282-x%29=0 ALIGN=MIDDLE  ALT=\"%282-x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Bx=0 ALIGN=MIDDLE  ALT=\"2%2Bx=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2 ALIGN=MIDDLE  ALT=\"x=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The roots to the above equation are <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2 ALIGN=MIDDLE  ALT=\"x=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " however, we must check each one to see what the slope is at these points.\r<BR>\n" );
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document.write( "At <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we have \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28dy%2Fdx%29=120%2A0%5E2-15%2A0%5E4=0 ALIGN=MIDDLE  ALT=\"%28dy%2Fdx%29=120%2A0%5E2-15%2A0%5E4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "At <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2 ALIGN=MIDDLE  ALT=\"x=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we have \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28dy%2Fdx%29=120%2A%28-2%29%5E2-15%2A%28-2%29%5E4=240 ALIGN=MIDDLE  ALT=\"%28dy%2Fdx%29=120%2A%28-2%29%5E2-15%2A%28-2%29%5E4=240\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "At <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we have \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28dy%2Fdx%29=120%2A%282%29%5E2-15%2A%282%29%5E4=240 ALIGN=MIDDLE  ALT=\"%28dy%2Fdx%29=120%2A%282%29%5E2-15%2A%282%29%5E4=240\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Therefore, the tangent line to the curve has the largest slope at \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2 ALIGN=MIDDLE  ALT=\"x=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and the value of the largest slope is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=240 ALIGN=MIDDLE  ALT=\"240\"   DISABLED_style= \"max-width:90%; height:auto;\" > at these points. \r<BR>\n" );
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document.write( " </SPAN>\n" );
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