document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=57281>Question 57281</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please Please Help!\r<BR>\n" );
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document.write( "Find all the points of inflection (concavity)\r<BR>\n" );
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document.write( "f(x)= x^4-18x^2+4 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #38971 by </B><A HREF=/tutors/aboutme.mpl?userid=funmath><B>funmath(2933)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=funmath><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=38971\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I am assuming that you are in Calculus.  Points of inflection are the points on a graph that the function changes concavity (either from up to down or from down to up.  You find the points of inflection by setting the second derivative equal to zero.<BR>\n" );
document.write( "Find all the points of inflection. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=+x%5E4-18x%5E2%2B4 ALIGN=MIDDLE  ALT=\"f%28x%29=+x%5E4-18x%5E2%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Take the first derivative.<BR>\n" );
document.write( "f'(x)=4x^3-36x  Take the derivative again, this is the second derivative.<BR>\n" );
document.write( "f\"(x)=12x^2-36  Set the derivative = 0, factor and solve.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=12x%5E2-36 ALIGN=MIDDLE  ALT=\"0=12x%5E2-36\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=12%28x%5E2-3%29 ALIGN=MIDDLE  ALT=\"0=12%28x%5E2-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%2F12=x%5E2-3 ALIGN=MIDDLE  ALT=\"0%2F12=x%5E2-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=x%5E2-3 ALIGN=MIDDLE  ALT=\"0=x%5E2-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3=x%5E2 ALIGN=MIDDLE  ALT=\"3=x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2B-sqrt%283%29=sqrt%28x%5E2%29 ALIGN=MIDDLE  ALT=\"%2B-sqrt%283%29=sqrt%28x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%283%29 ALIGN=MIDDLE  ALT=\"x=sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-sqrt%283%29 ALIGN=MIDDLE  ALT=\"x=-sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "These are the x values for the points of inflection, you have to plug these values into the original f(x) to find the y values.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28sqrt%283%29%29=%28sqrt%283%29%29%5E4-18%28sqrt%283%29%29%5E2%2B4 ALIGN=MIDDLE  ALT=\"f%28sqrt%283%29%29=%28sqrt%283%29%29%5E4-18%28sqrt%283%29%29%5E2%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28sqrt%283%29%29=9-18%283%29%2B4 ALIGN=MIDDLE  ALT=\"f%28sqrt%283%29%29=9-18%283%29%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28sqrt%283%29%29=9-54%2B4 ALIGN=MIDDLE  ALT=\"f%28sqrt%283%29%29=9-54%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28sqrt%283%29%29=-41 ALIGN=MIDDLE  ALT=\"f%28sqrt%283%29%29=-41\"   DISABLED_style= \"max-width:90%; height:auto;\" > This point of inflection is (sqrt(3) ,-41).<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-sqrt%283%29%29=%28-sqrt%283%29%29%5E4-18%28-sqrt%283%29%29%5E2%2B4 ALIGN=MIDDLE  ALT=\"f%28-sqrt%283%29%29=%28-sqrt%283%29%29%5E4-18%28-sqrt%283%29%29%5E2%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-sqrt%283%29%29=9-54%2B4 ALIGN=MIDDLE  ALT=\"f%28-sqrt%283%29%29=9-54%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-sqrt%283%29%29=-41 ALIGN=MIDDLE  ALT=\"f%28-sqrt%283%29%29=-41\"   DISABLED_style= \"max-width:90%; height:auto;\" >  This point of inflection is (-sqrt(3),-41)<BR>\n" );
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document.write( "When you look at the graph you realize this is resonable.  (I actually caught a mistake when I checked myself against the graph when I solved your problem.)  Here's the graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C200%2C-10%2C10%2C-80%2C6%2Cx%5E4-18x%5E2%2B4%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C200%2C-10%2C10%2C-80%2C6%2Cx%5E4-18x%5E2%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Can you tell about when the graph looks like a cup is turning upside down, then right side up?<BR>\n" );
document.write( "Happy Calculating!!! </SPAN>\n" );
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