document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1111276>Question 1111276</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the minimum slope of the line tangent to the curve y = x³-3x²+6x+3. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #726336 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=726336\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The slope of the line tangent to the curve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x%5E3-3x%5E2%2B6x%2B3 ALIGN=MIDDLE  ALT=\"y+=+x%5E3-3x%5E2%2B6x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "for each value 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;\" > is the value of the derivative,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx=3x%5E2-6x%2B5 ALIGN=MIDDLE  ALT=\"dy%2Fdx=3x%5E2-6x%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-4%2C6%2C-20%2C20%2C%0D%0Agraph%28300%2C300%2C-4%2C6%2C-20%2C20%2Cx%5E3-3x%5E2%2B6x%2B3%2C3.75x%2B3.5%2C6x-1%29%2C%0D%0Agreen%28circle%280.5%2C5.375%2C0.1%29%29%2Cblue%28circle%282%2C11%2C0.1%29%29%2C%0D%0Alocate%280.9%2C5%2Cblue%28slope=6%29%29%2Clocate%280.8%2C3%2Cblue%28at%29%29%2Clocate%281.5%2C3%2Cblue%28x=2%29%29%2C%0D%0Alocate%283%2C15.3%2Cgreen%28slope=3.75%29%29%2Clocate%283%2C13.3%2Cgreen%28at%29%29%2Clocate%283.7%2C13.3%2Cgreen%28x=0.5%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=slope=3x%5E2-6x%2B5 ALIGN=MIDDLE  ALT=\"slope=3x%5E2-6x%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a quadratic polynomial and graphs as a parabola.<BR>\n" );
document.write( "You are asked for the minimum value of that expression,<BR>\n" );
document.write( "and there are several ways to get to the result.<BR>\n" );
document.write( "Can you figure out what way your teacher would favor?<BR>\n" );
document.write( " <BR>\n" );
document.write( "Maybe your teacher expects you to find the minimum of<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=slope=3x%5E2-6x%2B5 ALIGN=MIDDLE  ALT=\"slope=3x%5E2-6x%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > by looking at<BR>\n" );
document.write( "zeros and changes of sign of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%28slope%29%2Fdx=6x-6 ALIGN=MIDDLE  ALT=\"d%28slope%29%2Fdx=6x-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "As <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%28slope%29%2Fdx%3C0 ALIGN=MIDDLE  ALT=\"d%28slope%29%2Fdx%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C1 ALIGN=MIDDLE  ALT=\"x%3C1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%28slope%29%2Fdx=0 ALIGN=MIDDLE  ALT=\"d%28slope%29%2Fdx=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%28slope%29%2Fdx%3E0 ALIGN=MIDDLE  ALT=\"d%28slope%29%2Fdx%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E1 ALIGN=MIDDLE  ALT=\"x%3E1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "there is a minimum at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The minimum slope, for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=slope=3%2A1%5E2-6%2A1%2B5=3-6%2B5=2 ALIGN=MIDDLE  ALT=\"slope=3%2A1%5E2-6%2A1%2B5=3-6%2B5=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "Alternately, if you memorized class information,<BR>\n" );
document.write( "you know that the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3E0 ALIGN=MIDDLE  ALT=\"a%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "has a minimum at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-b%29%2F%222+a%22 ALIGN=MIDDLE  ALT=\"x=%28-b%29%2F%222+a%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Maybe your teacher expects you to do that,<BR>\n" );
document.write( "find that the minimum is at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-%28-6%29%29%2F%282%2A3%29=6%2F6=1 ALIGN=MIDDLE  ALT=\"x=%28-%28-6%29%29%2F%282%2A3%29=6%2F6=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and calculate that for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=slope=3%2A1%5E2-6%2A1%2B5=3-6%2B5=2 ALIGN=MIDDLE  ALT=\"slope=3%2A1%5E2-6%2A1%2B5=3-6%2B5=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "You could just use your algebra knowledge (no memorization)<BR>\n" );
document.write( "and \"complete the square\" like this<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=slope=3x%5E2-6x%2B5 ALIGN=MIDDLE  ALT=\"slope=3x%5E2-6x%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=slope%2F3=x%5E2-2x%2B5%2F3 ALIGN=MIDDLE  ALT=\"slope%2F3=x%5E2-2x%2B5%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=slope%2F3=%28x-1%29%5E2%2B5%2F3-1 ALIGN=MIDDLE  ALT=\"slope%2F3=%28x-1%29%5E2%2B5%2F3-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=slope%2F3=%28x-1%29%5E2%2B2%2F3 ALIGN=MIDDLE  ALT=\"slope%2F3=%28x-1%29%5E2%2B2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=slope=3%28%28x-1%29%5E2%2B2%2F3%29 ALIGN=MIDDLE  ALT=\"slope=3%28%28x-1%29%5E2%2B2%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=slope=3%28x-1%29%5E2%2B2%29 ALIGN=MIDDLE  ALT=\"slope=3%28x-1%29%5E2%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "As <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%5E2%3E=0 ALIGN=MIDDLE  ALT=\"%28x-1%29%5E2%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=slope%3E=3 ALIGN=MIDDLE  ALT=\"slope%3E=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );