document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=401556>Question 401556</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Topic: Maxima and Minima for functions of 2 variables:\r<BR>\n" );
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document.write( "Question: Find the critical points of the question below by determining whether it is a saddle point, relative maximum or minimum:<BR>\n" );
document.write( "f(x,y) = xy(1-x-y)\r<BR>\n" );
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document.write( "Thank-you for your assistance. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #284236 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=284236\">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=f%28x%2Cy%29+=+xy%281-x-y%29+++=+xy+-+x%5E2y-xy%5E2 ALIGN=MIDDLE  ALT=\"f%28x%2Cy%29+=+xy%281-x-y%29+++=+xy+-+x%5E2y-xy%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5Bx%5D+=+y-2xy+-+y%5E2+=+y%281-2x-y%29+=+0 ALIGN=MIDDLE  ALT=\"f%5Bx%5D+=+y-2xy+-+y%5E2+=+y%281-2x-y%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5By%5D+=+x-2xy+-+x%5E2+=+x%281-x-2y%29+=+0 ALIGN=MIDDLE  ALT=\"f%5By%5D+=+x-2xy+-+x%5E2+=+x%281-x-2y%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Solving the preceding system above, we get the following c.p.'s<BR>\n" );
document.write( "(i) (0,0)<BR>\n" );
document.write( "{ii) (1,0)<BR>\n" );
document.write( "(iii) (0,1)<BR>\n" );
document.write( "(iv) (1/3, 1/3)<BR>\n" );
document.write( "Finding the 2nd order partial derivatives:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5Bxx%5D+=+-2y ALIGN=MIDDLE  ALT=\"f%5Bxx%5D+=+-2y\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5Byy%5D+=+-2x ALIGN=MIDDLE  ALT=\"f%5Byy%5D+=+-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5Bxy%5D+=+1-2x+-+2y+=+f%5Byx%5D ALIGN=MIDDLE  ALT=\"f%5Bxy%5D+=+1-2x+-+2y+=+f%5Byx%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(i) At (0,0), <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5Bxx%5D+=+0 ALIGN=MIDDLE  ALT=\"f%5Bxx%5D+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5Byy%5D+=+0 ALIGN=MIDDLE  ALT=\"f%5Byy%5D+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5Bxy%5D+=+1 ALIGN=MIDDLE  ALT=\"f%5Bxy%5D+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "D = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5Bxx%5D%2A+f%5Byy%5D+-+%28f%5Bxy%5D%29%5E2+=+0+%2A+0+-+1+=+-1+%3C+0 ALIGN=MIDDLE  ALT=\"f%5Bxx%5D%2A+f%5Byy%5D+-+%28f%5Bxy%5D%29%5E2+=+0+%2A+0+-+1+=+-1+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Therefore there is a saddle point at (0,0).\r<BR>\n" );
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document.write( "(ii) At (1,0), <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5Bxx%5D+=+0 ALIGN=MIDDLE  ALT=\"f%5Bxx%5D+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5Byy%5D+=+-2 ALIGN=MIDDLE  ALT=\"f%5Byy%5D+=+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5Bxy%5D+=+-1 ALIGN=MIDDLE  ALT=\"f%5Bxy%5D+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "D = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5Bxx%5D%2A+f%5Byy%5D+-+%28f%5Bxy%5D%29%5E2+=+0+%2A+-2++-+1+=+-1+%3C+0 ALIGN=MIDDLE  ALT=\"f%5Bxx%5D%2A+f%5Byy%5D+-+%28f%5Bxy%5D%29%5E2+=+0+%2A+-2++-+1+=+-1+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Therefore there is a saddle point at (1,0)\r<BR>\n" );
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document.write( "(iii) At (0,1), <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5Bxx%5D+=+-2 ALIGN=MIDDLE  ALT=\"f%5Bxx%5D+=+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5Byy%5D+=+0 ALIGN=MIDDLE  ALT=\"f%5Byy%5D+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5Bxy%5D+=+-1 ALIGN=MIDDLE  ALT=\"f%5Bxy%5D+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "D = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5Bxx%5D%2A+f%5Byy%5D+-+%28f%5Bxy%5D%29%5E2+=+-2+%2A+0+-+1+=+-1+%3C+0 ALIGN=MIDDLE  ALT=\"f%5Bxx%5D%2A+f%5Byy%5D+-+%28f%5Bxy%5D%29%5E2+=+-2+%2A+0+-+1+=+-1+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Therefore there is saddle point at (0,1).\r<BR>\n" );
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document.write( "(iv) At (1/3, 1/3), <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5Bxx%5D=+-2%2F3 ALIGN=MIDDLE  ALT=\"f%5Bxx%5D=+-2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5Byy%5D+=+-2%2F3 ALIGN=MIDDLE  ALT=\"f%5Byy%5D+=+-2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5Bxy%5D+=+-1%2F3 ALIGN=MIDDLE  ALT=\"f%5Bxy%5D+=+-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "D = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5Bxx%5D%2A+f%5Byy%5D+-+%28f%5Bxy%5D%29%5E2+=+%28-2%2F3%29%5E2+-+1%2F9+=+4%2F9-1%2F9+=+1%2F3+%3E+0 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Therefore there is a relative maximum at (1/3, 1/3).\r<BR>\n" );
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document.write( " </SPAN>\n" );
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