document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=463056>Question 463056</A>: <I><SPAN STYLE=\"word-break: break-all;\"> (a) Find the value of m for which the line y=mx-3 is a tangent to the curve y=x+1/x and find the x-coordinate of the point at which this tangent touches the curve.\r<BR>\n" );
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document.write( "(b) Find the values of c and d for which {x:-5<x<3} is the solution set of x^2+cx<d.\r<BR>\n" );
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document.write( "ps. I have the answers but i do not know how to derive to the answer. Your help will be appreciated. Thank you:)\r<BR>\n" );
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document.write( "answers: (a)m=-5/4, x=-2/3               (b)c=2, d=15 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #317389 by </B><A HREF=/tutors/aboutme.mpl?userid=htmentor><B>htmentor(1343)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=htmentor><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=htmentor><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=317389\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> (a) Find the value of m for which the line y=mx-3 is a tangent to the curve y=x+1/x and find the x-coordinate of the point at which this tangent touches the curve.\r<BR>\n" );
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document.write( "(b) Find the values of c and d for which {x:-5<x<3} is the solution set of x^2+cx<d.\r<BR>\n" );
document.write( "\n" );
document.write( "ps. I have the answers but i do not know how to derive to the answer. Your help will be appreciated. Thank you:)\r<BR>\n" );
document.write( "\n" );
document.write( "answers: (a)m=-5/4, x=-2/3               (b)c=2, d=15<BR>\n" );
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document.write( "(a) The derivative of the curve at a point (x0,y0) is the slope of the tangent line at that point.<BR>\n" );
document.write( "The curve and its tangent line intersect at the point (x0,y0).  <BR>\n" );
document.write( "The equation for the curve is y = x + 1/x<BR>\n" );
document.write( "The equation for the tangent line is y = mx - 3<BR>\n" );
document.write( "y = x + 1/x<BR>\n" );
document.write( "y = mx - 3<BR>\n" );
document.write( "So we can equate the RHS's:<BR>\n" );
document.write( "x + 1/x = mx - 3 [1]<BR>\n" );
document.write( "But the slope of the tangent line, dy/dx = m = 1 - 1/x^2 [2]<BR>\n" );
document.write( "Solve for m in [1] and substitute into [2]<BR>\n" );
document.write( "x + 1/x + 3 = mx <BR>\n" );
document.write( "1 + 1/x^2 + 3/x = m<BR>\n" );
document.write( "1 + 1/x^2 + 3/x = 1 - 1/x^2<BR>\n" );
document.write( "2/x^2 + 3/x = 0<BR>\n" );
document.write( "2 + 3x = 0<BR>\n" );
document.write( "So x0 = -2/3<BR>\n" );
document.write( "Since the point (x0,y0) lies on the curve y = x + 1/x, we have y0 = -2/3 + 1/(-2/3)<BR>\n" );
document.write( "This gives y0 = -13/6<BR>\n" );
document.write( "So the slope of the tangent line is (y0 - b)/(x0 - 0) = (-13/6 + 3)/(-3/2) = -5/4 </SPAN>\n" );
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