document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1179606>Question 1179606</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I'm trying to find where the tangent line to a function is horizontal. The equation is y = e^(-sqrt(3)x)cos(x). I got the derivative, (-e^(-sqrt(3)x))(sqrt(3)cos(x)+sin(x)), and I set each factor equal to zero. The first factor has no solutions, but the solution for the second one is -pi/3. I then substituted -pi/3 into the original equation to find the y coordinate for the point where the tangent is horizontal. (I got (e^(sqrt(3)pi/3)/2 for the y coordinate).<BR>\n" );
document.write( "Is this correct?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #809179 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</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=809179\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "you found <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D=-pi%2F3 ALIGN=MIDDLE  ALT=\"x%5B1%5D=-pi%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and got <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D=%28e%5E%28sqrt%283%29%28pi%2F3%29%29%29%2F2 ALIGN=MIDDLE  ALT=\"y%5B1%5D=%28e%5E%28sqrt%283%29%28pi%2F3%29%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( " which is a point of tangency\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Secondly, find the slope of the tangent line, which is the derivative of the function, evaluated at the point: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f ALIGN=MIDDLE  ALT=\"f\"   DISABLED_style= \"max-width:90%; height:auto;\" >′<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   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=m=+-e%5E%28-sqrt%283%29%28-pi%2F3%29%29+%28sin%28-pi%2F3%29+%2B+sqrt%283%29+cos%28-pi%2F3%29%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=m=0 ALIGN=MIDDLE  ALT=\"m=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > => if so, we have horizontal line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=a ALIGN=MIDDLE  ALT=\"y=a\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "in your case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=y%5B1%5D ALIGN=MIDDLE  ALT=\"y=y%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > or \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28e%5E%28sqrt%283%29%28pi%2F3%29%29%29%2F2 ALIGN=MIDDLE  ALT=\"y=%28e%5E%28sqrt%283%29%28pi%2F3%29%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >-> tangent\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "check if that is tangent line\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2Ce%5E%28-sqrt%283%29x%29%2Acos%28x%29%2C%28e%5E%28sqrt%283%29pi%2F3%29%29%2F2%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );