document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=822167>Question 822167</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find two numbers whos sum is 21 and whose product is a maximum </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #494646 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=494646\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let us assume one of the numbers is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >. As the sum of the two numbers is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21 ALIGN=MIDDLE  ALT=\"21\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the other number is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21-x ALIGN=MIDDLE  ALT=\"21-x\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "\n" );
document.write( "The product of the two numbers is a function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+x%2A%2821-x%29+=+21x-+x%5E2 ALIGN=MIDDLE  ALT=\"f%28x%29+=+x%2A%2821-x%29+=+21x-+x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "\n" );
document.write( "Now we need to find the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=maximum ALIGN=MIDDLE  ALT=\"maximum\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+value ALIGN=MIDDLE  ALT=\"+value\"   DISABLED_style= \"max-width:90%; height:auto;\" > of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
document.write( "\n" );
document.write( "For this we need the derivative of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and have to equate it to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "\n" );
document.write( "f’<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%29+=+21+%96+2x+=+0 ALIGN=MIDDLE  ALT=\"%28x%29+=+21+%96+2x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "=> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10.5+%96+x+=0 ALIGN=MIDDLE  ALT=\"10.5+%96+x+=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "=> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=10.5 ALIGN=MIDDLE  ALT=\"x+=10.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Now f’’<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%29+=+-2 ALIGN=MIDDLE  ALT=\"%28x%29+=+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is negative at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=10.5 ALIGN=MIDDLE  ALT=\"x=10.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f+%2810.5%29 ALIGN=MIDDLE  ALT=\"f+%2810.5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is truly the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=maximum+value ALIGN=MIDDLE  ALT=\"maximum+value\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=10.5 ALIGN=MIDDLE  ALT=\"x=10.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the first number is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10.5 ALIGN=MIDDLE  ALT=\"10.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the second number is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21-10.5+=+10.5 ALIGN=MIDDLE  ALT=\"21-10.5+=+10.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "So the two required numbers are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10.5 ALIGN=MIDDLE  ALT=\"10.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10.5 ALIGN=MIDDLE  ALT=\"10.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+600%2C+600%2C+-15%2C+35%2C+-10%2C+115%2C+-x%5E2%2B21x%29+ ALIGN=MIDDLE  ALT=\"+graph%28+600%2C+600%2C+-15%2C+35%2C+-10%2C+115%2C+-x%5E2%2B21x%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );