document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1026930>Question 1026930</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The temperature of a point (x,y) in the plane is given by the expression x^2 + y^2 - 4x + 2y. What is the temperature of the coldest point in the plane?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #642210 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=642210\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F+=++x%5E2+%2B+y%5E2+-+4x+%2B+2y ALIGN=MIDDLE  ALT=\"F+=++x%5E2+%2B+y%5E2+-+4x+%2B+2y\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Finding the critical points of F: <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F%5Bx%5D+=++2x+-+4+=+0 ALIGN=MIDDLE  ALT=\"F%5Bx%5D+=++2x+-+4+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >==> x = 2, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F%5By%5D+=++2y+%2B+2+=+0 ALIGN=MIDDLE  ALT=\"F%5By%5D+=++2y+%2B+2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >==> y = -1.<BR>\n" );
document.write( "==> critical point is (2,-1).<BR>\n" );
document.write( "Also, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F%5Bxx%5D+=+2+%3E+0 ALIGN=MIDDLE  ALT=\"F%5Bxx%5D+=+2+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F%5Byy%5D+=+2+%3E+0 ALIGN=MIDDLE  ALT=\"F%5Byy%5D+=+2+%3E+0\"   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+=+0 ALIGN=MIDDLE  ALT=\"F%5Bxy%5D+=+F%5Byx%5D+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Implement the 2nd derivative test for two variables:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F%5Bxx%5D%2AF%5Byy%5D+-+%28F%5Bxy%5D%29%5E2+=+2%2A2+-+0%5E2+=+4+%3E+0 ALIGN=MIDDLE  ALT=\"F%5Bxx%5D%2AF%5Byy%5D+-+%28F%5Bxy%5D%29%5E2+=+2%2A2+-+0%5E2+=+4+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "==> There is local min at (2,-1).  Since it is the only critical point in the domain of the function (which is infinite open), it is also an absolute minimum.<BR>\n" );
document.write( "The temperature of the coldest point is thus <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E2+%2B+%28-1%29%5E2+-+4%2A2+%2B+2%2A%28-1%29+=+-5 ALIGN=MIDDLE  ALT=\"2%5E2+%2B+%28-1%29%5E2+-+4%2A2+%2B+2%2A%28-1%29+=+-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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