document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1033076>Question 1033076</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the point of inflexion on the<BR>\n" );
document.write( "curve y=xln(x)-x^2.\r<BR>\n" );
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document.write( "I found the derivative: y'= ln(x+1) -2x<BR>\n" );
document.write( "But I can't find the point of inflexion because I can't solve for y'=0<BR>\n" );
document.write( "Please Help !!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #647698 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=647698\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=xlnx-x%5E2 ALIGN=MIDDLE  ALT=\"y=xlnx-x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >  ==> y' = lnx + 1 - 2x  ==> y\" = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fx-2 ALIGN=MIDDLE  ALT=\"1%2Fx-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The critical values of the 2nd derivative are x = 0 and x = 1/2, but the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=xlnx-x%5E2 ALIGN=MIDDLE  ALT=\"y=xlnx-x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is undefined at x = 0, so there is no inflection point there.\r<BR>\n" );
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document.write( "Now in (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-infinity ALIGN=MIDDLE  ALT=\"-infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >, 0), y\" < 0.<BR>\n" );
document.write( "In (0, 1/2), y\" > 0, and <BR>\n" );
document.write( "at (1/2, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >), y\" < 0.<BR>\n" );
document.write( "(At x = 1/2, y\" = 0.)\r<BR>\n" );
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document.write( "Hence there is only one inflection point at (1/2, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-ln2%2F2-1%2F4 ALIGN=MIDDLE  ALT=\"-ln2%2F2-1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >).  (Even though there is a change in the direction of concavity from the interval (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-infinity ALIGN=MIDDLE  ALT=\"-infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >, 0) to (0, 1/2).)\r<BR>\n" );
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