document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=534799>Question 534799</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Identify all Maximums Points, Minimum Points, and Points of Inflections for the function (x+3)/(x^2-16).<BR>\n" );
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document.write( "I am not sure. I am thinking that ff there is no point at which the first derivative is equal to 0, then it is impossible for there to be a point at which both the second and first derivative are equal to 0, making the answer for all three points (Maximum, Minimum, and Points of Inflection) none. \r<BR>\n" );
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document.write( "My domain is all values of x except x=-2 & x=2. My range is (-infinity, infinity).\r<BR>\n" );
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document.write( "Thank you! :) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #351694 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=351694\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If it's really <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x%2B3%29%2F%28x%5E2-16%29=%28x%2B3%29%2F%28x%2B4%29%28x-4%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%28x%2B3%29%2F%28x%5E2-16%29=%28x%2B3%29%2F%28x%2B4%29%28x-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "the points where it does not exist are x=-4 and x=4, and all other x are in the domain.<BR>\n" );
document.write( "I agree that the range is (-infinity, infinity).<BR>\n" );
document.write( "If that's really the function, it changes sign 3 times (at -4, -3 an 3), with vertical asymptotes at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-4 ALIGN=MIDDLE  ALT=\"x=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and a zero at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3 ALIGN=MIDDLE  ALT=\"x=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "There is no minimum or maximum. The first derivative is always negative, so the function decreases continuously in each of the 3 regions where it exists, going from positive infinity at the right of vertical asymptote <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-4 ALIGN=MIDDLE  ALT=\"x=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "down to negative infinity on the left of vertical asymptote <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "To hug those vertical asymptotes like that, the function has to go  from concave upwards to concave downwards. The second derivative must have an inflection point somewhere between them. I calculated it, but I may have made a mistake. I got <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%5E3%2B18x%5E2%2B96x%2B96%29%2F%28x%5E2-16%29%5E3 ALIGN=MIDDLE  ALT=\"%282x%5E3%2B18x%5E2%2B96x%2B96%29%2F%28x%5E2-16%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which is zero at a point near <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1.25 ALIGN=MIDDLE  ALT=\"x=-1.25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You do not need both derivatives to be zero to have an inflection point.<BR>\n" );
document.write( "If they are both zero, it is a saddle point, an inflection point where the slope is zero. An example of a function with an inflection point where the first derivative is not zero is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28x%29 ALIGN=MIDDLE  ALT=\"tan%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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