document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=446615>Question 446615</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If a , b and c are three sides of a triangle with perimeter 1, then is bc + ca + ab less than or equal to 1/3? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #307708 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=307708\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We show that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ab+%2B+bc+%2B+ca+%3C=+1%2F3 ALIGN=MIDDLE  ALT=\"ab+%2B+bc+%2B+ca+%3C=+1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "We want to optimize ab + bc + ca subject to the constraint a + b + c = 1.\r<BR>\n" );
document.write( "\n" );
document.write( "With Lagrange multiplier <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha ALIGN=MIDDLE  ALT=\"alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" > in mind, define the function<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F%28a%2Cb%2Cc%29+=+ab+%2B+bc+%2B+ca++%2B+alpha%2A%28a%2Bb%2Bc+-+1%29 ALIGN=MIDDLE  ALT=\"F%28a%2Cb%2Cc%29+=+ab+%2B+bc+%2B+ca++%2B+alpha%2A%28a%2Bb%2Bc+-+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F%5Ba%5D+=+b%2Bc%2B+alpha+=+0 ALIGN=MIDDLE  ALT=\"F%5Ba%5D+=+b%2Bc%2B+alpha+=+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%5Bb%5D+=+a%2Bc%2B+alpha+=+0 ALIGN=MIDDLE  ALT=\"F%5Bb%5D+=+a%2Bc%2B+alpha+=+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%5Bc%5D+=+a%2Bb%2B+alpha+=+0 ALIGN=MIDDLE  ALT=\"F%5Bc%5D+=+a%2Bb%2B+alpha+=+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%5Balpha%5D+=+a%2Bb%2Bc+-+1+=+0 ALIGN=MIDDLE  ALT=\"F%5Balpha%5D+=+a%2Bb%2Bc+-+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "\n" );
document.write( "Solving the system of equations first for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha ALIGN=MIDDLE  ALT=\"alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we obtain <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha+=+-2%2F3 ALIGN=MIDDLE  ALT=\"alpha+=+-2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The system then becomes <BR>\n" );
document.write( "b+c = 2/3,<BR>\n" );
document.write( "a+c = 2/3, and <BR>\n" );
document.write( "a+b = 2/3.\r<BR>\n" );
document.write( "\n" );
document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+b+=+c+=+1%2F3 ALIGN=MIDDLE  ALT=\"a+=+b+=+c+=+1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "These values of a,b, and c would give optimize ab + bc + ca, we just don't know whether its a max or min.  When <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+b+=+c+=+1%2F3 ALIGN=MIDDLE  ALT=\"a+=+b+=+c+=+1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ab+%2B+bc+%2B+ca+=+1%2F3 ALIGN=MIDDLE  ALT=\"ab+%2B+bc+%2B+ca+=+1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "\n" );
document.write( "Hence we compare the value of ab + bc + ca when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+1%2F4 ALIGN=MIDDLE  ALT=\"a+=+1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+1%2F4 ALIGN=MIDDLE  ALT=\"b+=+1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=+1%2F2 ALIGN=MIDDLE  ALT=\"c+=+1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which gives <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ab+%2B+bc+%2B+ca+=+1%2F16+%2B+2%2F16+%2B+2%2F16+=+5%2F16+%3C+1%2F3 ALIGN=MIDDLE  ALT=\"ab+%2B+bc+%2B+ca+=+1%2F16+%2B+2%2F16+%2B+2%2F16+=+5%2F16+%3C+1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Hence <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+b+=+c+=+1%2F3 ALIGN=MIDDLE  ALT=\"a+=+b+=+c+=+1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > maximizes ab + bc + ca, and <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ab+%2B+bc+%2B+ca+%3C=+1%2F3 ALIGN=MIDDLE  ALT=\"ab+%2B+bc+%2B+ca+%3C=+1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );