document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1088908>Question 1088908</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider all the points in the plane that solve the equation x^2 + 2y^2 = 16. Find the maximum value of the product xy on this graph. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #703230 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=703230\">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=x%5E2+%2B+2y%5E2+=+16 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+2y%5E2+=+16\"   DISABLED_style= \"max-width:90%; height:auto;\" > represents an ellipse centered at (0,0),<BR>\n" );
document.write( "which is symmetrical with respect the the x- and y-axes,<BR>\n" );
document.write( "and symmetrical with respect to the origin:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-4.4%2C4.4%2C-4.4%2C4.4%2C%0D%0Agrid%280%29%2Carc%280%2C0%2C8%2C4sqrt%282%29%2C0%2C360%29%2C%0D%0Acircle%282.83%2C2%2C0.1%29%2Ccircle%28-2.83%2C-2%2C0.1%29%2C%0D%0Alocate%283%2C2.3%2CM%29%2Clocate%28-3.2%2C-2%2CP%29+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "There will be a maximum for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Ay ALIGN=MIDDLE  ALT=\"x%2Ay\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the first quadrant, at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M%28x%5BM%5D%2Cy%5BM%5D%29 ALIGN=MIDDLE  ALT=\"M%28x%5BM%5D%2Cy%5BM%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and another one at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28-x%5BM%5D%2C-y%5BM%5D%29 ALIGN=MIDDLE  ALT=\"P%28-x%5BM%5D%2C-y%5BM%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , in quadrant III.<BR>\n" );
document.write( "All we have to do is find two positive numbers, the coordinates of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M ALIGN=MIDDLE  ALT=\"M\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+2y%5E2+=+16 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+2y%5E2+=+16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+=+16+-+2y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2+=+16+-+2y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+y%5E2=+%2816+-+2y%5E2%29y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2+y%5E2=+%2816+-+2y%5E2%29y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28xy%29%5E2=+16y%5E2+-+2y%5E4 ALIGN=MIDDLE  ALT=\"%28xy%29%5E2=+16y%5E2+-+2y%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We are looking for a value <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=y%5BM%5D%3E0 ALIGN=MIDDLE  ALT=\"y=y%5BM%5D%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > that makes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16y%5E2+-+2y%5E4 ALIGN=MIDDLE  ALT=\"16y%5E2+-+2y%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > maximum.<BR>\n" );
document.write( "If we say <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=y%5E2 ALIGN=MIDDLE  ALT=\"t=y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the expression becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16t-2t%5E2 ALIGN=MIDDLE  ALT=\"16t-2t%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "a quadratic polynomial/function:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-1%2C9%2C-10%2C40%2C16x-2x%5E2%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-1%2C9%2C-10%2C40%2C16x-2x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If you like applying formulas (or if your teacher likes to see formulas),<BR>\n" );
document.write( "you may have been told in class that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"f%28x%29=ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > is maximum when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-b%29%2F%222+a%22 ALIGN=MIDDLE  ALT=\"x=%28-b%29%2F%222+a%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In this case, it means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16t-2t%5E2 ALIGN=MIDDLE  ALT=\"16t-2t%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is maximum<BR>\n" );
document.write( "when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=%28-16%29%2F%282%28-2%29%29=%28-16%29%2F%28-4%29=4 ALIGN=MIDDLE  ALT=\"t=%28-16%29%2F%282%28-2%29%29=%28-16%29%2F%28-4%29=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "Otherwise, you may realize that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16t-2t%5E2=-2%28t%5E2-8t%2B16-16%29=-2%28t%5E2-8t%2B16%29%2B32=-2%28t-4%29%5E2%2B32%3C=32 ALIGN=MIDDLE  ALT=\"16t-2t%5E2=-2%28t%5E2-8t%2B16-16%29=-2%28t%5E2-8t%2B16%29%2B32=-2%28t-4%29%5E2%2B32%3C=32\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That tells you that the maximum of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28xy%29%5E2 ALIGN=MIDDLE  ALT=\"%28xy%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32 ALIGN=MIDDLE  ALT=\"32\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "meaning that the maximum of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xy ALIGN=MIDDLE  ALT=\"xy\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2832%29=highlight%284sqrt%282%29%29 ALIGN=MIDDLE  ALT=\"sqrt%2832%29=highlight%284sqrt%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "If you continue working with formulas, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=4 ALIGN=MIDDLE  ALT=\"t=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >  gives you <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5BM%5D%5E2=4 ALIGN=MIDDLE  ALT=\"y%5BM%5D%5E2=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5BM%5D=2 ALIGN=MIDDLE  ALT=\"y%5BM%5D=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5BM%5D%5E2+=+16+-+2y%5BM%5D%5E2=16-2%2A4=16-8=8 ALIGN=MIDDLE  ALT=\"x%5BM%5D%5E2+=+16+-+2y%5BM%5D%5E2=16-2%2A4=16-8=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5BM%5D=sqrt%288%29=2sqrt%282%29 ALIGN=MIDDLE  ALT=\"x%5BM%5D=sqrt%288%29=2sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Multiplying, you get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5BM%5Dy%5BM%5D=%282sqrt%282%29%29%2A2=highlight%284sqrt%282%29%29=%28-x%5BM%5D%29%28-y%5BM%5D%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( ". </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );