document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1035337>Question 1035337</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the max/min  point for the following function:\r<BR>\n" );
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document.write( "f (x,y) = x^3 + y^3 + 6xy </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #650007 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=650007\">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%5Bx%5D+=+3x%5E2%2B6y ALIGN=MIDDLE  ALT=\"f%28x%2Cy%29%5Bx%5D+=+3x%5E2%2B6y\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%2Cy%29%5By%5D+=+3y%5E2%2B6x ALIGN=MIDDLE  ALT=\"f%28x%2Cy%29%5By%5D+=+3y%5E2%2B6x\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Also, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5Bxx%5D+=+6x ALIGN=MIDDLE  ALT=\"f%5Bxx%5D+=+6x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5Byy%5D+=+6y ALIGN=MIDDLE  ALT=\"f%5Byy%5D+=+6y\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5Bxy%5D=+f%5Byx%5D+=+6 ALIGN=MIDDLE  ALT=\"f%5Bxy%5D=+f%5Byx%5D+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Use the Second partial derivative test.\r<BR>\n" );
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document.write( "Set the partial derivatives above to 0 and solve for x and y.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B6y+=+0 ALIGN=MIDDLE  ALT=\"3x%5E2%2B6y+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+3y%5E2%2B6x+=+0 ALIGN=MIDDLE  ALT=\"+3y%5E2%2B6x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "This system of equations yield two points as its solutions:  (0,0) , (-2,-2).  (Verify!!)\r<BR>\n" );
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document.write( "Now find the determinant of the hessian for each point.\r<BR>\n" );
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document.write( "For (0,0): <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5Bxx%5D%2Af%5Byy%5D+-+%28f%5Bxy%5D%29%5E2+=+0%2A0-6%5E2+=+-36+%3C+0 ALIGN=MIDDLE  ALT=\"f%5Bxx%5D%2Af%5Byy%5D+-+%28f%5Bxy%5D%29%5E2+=+0%2A0-6%5E2+=+-36+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "This implies that the point (0,0) is a saddle point (neither a local max nor a local min).\r<BR>\n" );
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document.write( "For (-2,-2):  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5Bxx%5D%2Af%5Byy%5D+-+%28f%5Bxy%5D%29%5E2+=+-12%2A-12-6%5E2+=+144-36+%3E+0 ALIGN=MIDDLE  ALT=\"f%5Bxx%5D%2Af%5Byy%5D+-+%28f%5Bxy%5D%29%5E2+=+-12%2A-12-6%5E2+=+144-36+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5Bxx%5D+%3C+0 ALIGN=MIDDLE  ALT=\"f%5Bxx%5D+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, it follows that (-2,-2) is a local maximum.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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