document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1082928>Question 1082928</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the equation of the tangent line to the curve, y = x^3 - 3x^2 + 5x = 2 that has the least slope.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #697015 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=697015\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ x^3\ -\ 3x^2\ +\ 5x\ =\ 2\">\r<BR>\n" );
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document.write( "doesn't make any sense, so I figure you either meant +2 and you failed to hold the shift key when you pressed the +/= key, or you meant -2 and you simply pressed the +/= key instead of the _/- key.  Either scenario being equally likely, I'm going to do the problem for the following function:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  y\ =\ x^3\ -\ 3x^2\ +\ 5x\ +\ a\">\r<BR>\n" );
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document.write( "Where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\"> is an arbitrary real number constant.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  \frac{dy}{dx}\ =\ 3x^2\ -\ 6x\ +\ 5\">\r<BR>\n" );
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document.write( "The minimum slope tangent to the original function will occur at the point where the first derivative has a minimum.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  \frac{d^2y}{dx^2}\ =\ 6x\ -\ 6\">\r<BR>\n" );
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document.write( "Which has a zero at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ 1\">\r<BR>\n" );
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document.write( "For the original function\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  f(1)\ =\ 3\ +\ a\">\r<BR>\n" );
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document.write( "Hence, the point of tangency is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (1,\,3\,+\,a)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  f'(1)\ =\ 2\">\r<BR>\n" );
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document.write( "So you want an equation of a line that passes through the point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (1,\,3\,+\,a)\"> with a slope of 2.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ -\ (3\ +\ a)\ =\ 2(x\ -\ 1)\">\r<BR>\n" );
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document.write( "Once you figure out whether <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\ =\ 2\"> or <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\ =\ -2\">,\r<BR>\n" );
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document.write( "Then you can complete the problem.\r<BR>\n" );
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document.write( "By the way, the question was also worded incorrectly in the first place.  You cannot find THE equation of a line.  You can only find AN equation of a line.<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" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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