document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=862385>Question 862385</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the intervals of increase and decrease. Use the first derivatives test to find the local maxima and minima. Graph.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x%5E2-1%29%2F%28x%5E2%2B1%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%28x%5E2-1%29%2F%28x%5E2%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Derivative is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%2F%28x%5E2%2B1%29%5E2 ALIGN=MIDDLE  ALT=\"4x%2F%28x%5E2%2B1%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > I think </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #519643 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=519643\">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( "Nope.  Derivative is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{dy}{dx}\ =\ \frac{2x}{x^2\ +\ 1}\ +\ \frac{2x(x^2\ -\ 1)}{(x^2\ +\ 1)^2}\">\r<BR>\n" );
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document.write( "You need to use a combination of the quotient rule and the chain rule.\r<BR>\n" );
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document.write( "1st derivative is equal to zero only when x = 0, hence the only extremum is at x = 0.\r<BR>\n" );
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document.write( "Second derivative:\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}\ =\ \frac{2}{x^2\ +\ 1}\ -\ \frac{8x^2}{(x^2\ +\ 1)^2}\ -\ \frac{2(x^2\ -\ 1)^2}{(x^2\ +\ 1)^2}\ +\ \frac{8x^2(x^2\ -\ 1)^2}{(x^2\ +\ 1)^3}\">\r<BR>\n" );
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document.write( "Second derivative has the value 4 (> 0) at x = 0, hence the extremum is a minimum.\r<BR>\n" );
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document.write( "The denominator of the original function has no real zeros, hence there are no vertical asymptotes to the graph of this function.\r<BR>\n" );
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document.write( "I'll leave the graphing to you, but it sort of looks like a side view of an \"innie\" belly button with a minimum at (0,-1) and asymptotic to y = 1 as x increases or decreases without bound.\r<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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