document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=461593>Question 461593</A>: <I><SPAN STYLE=\"word-break: break-all;\"> prove that for x,y,z>=0, x^2+y^2+z^2>=xy+yz+zx </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #316526 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=316526\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Consider the vectors (x, y, z) and ( <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fsqrt%283%29 ALIGN=MIDDLE  ALT=\"1%2Fsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fsqrt%283%29 ALIGN=MIDDLE  ALT=\"1%2Fsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fsqrt%283%29 ALIGN=MIDDLE  ALT=\"1%2Fsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >).  <BR>\n" );
document.write( "Then direct application of the Cauchy-Schwartz Inequality gives:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2+%2B+y%5E2+%2B+z%5E2%29%2A%28%281%2Fsqrt%283%29%29%5E2+%2B%281%2Fsqrt%283%29%29%5E2+%2B+%281%2Fsqrt%283%29%29%5E2%29+%3E=+%28x%2Fsqrt%283%29+%2B+y%2Fsqrt%283%29+%2B+z%2Fsqrt%283%29%29%5E2 ALIGN=MIDDLE   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=%28x%5E2+%2B+y%5E2+%2B+z%5E2%29%2A1+%3E=+%281%2F3%29%28x%2By%2Bz%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%5E2+%2B+y%5E2+%2B+z%5E2%29%2A1+%3E=+%281%2F3%29%28x%2By%2Bz%29%5E2\"   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=3x%5E2+%2B+3y%5E2+%2B+3z%5E2+%3E=+x%5E2+%2B+y%5E2+%2B+z%5E2++%2B+2xy+%2B+2xz+%2B+2yz ALIGN=MIDDLE  ALT=\"3x%5E2+%2B+3y%5E2+%2B+3z%5E2+%3E=+x%5E2+%2B+y%5E2+%2B+z%5E2++%2B+2xy+%2B+2xz+%2B+2yz\"   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=2x%5E2+%2B2y%5E2+%2B+2z%5E2+%3E=++2xy+%2B+2xz+%2B+2yz ALIGN=MIDDLE  ALT=\"2x%5E2+%2B2y%5E2+%2B+2z%5E2+%3E=++2xy+%2B+2xz+%2B+2yz\"   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+%2By%5E2+%2B+z%5E2+%3E=++xy+%2B+xz+%2B+yz ALIGN=MIDDLE  ALT=\"x%5E2+%2By%5E2+%2B+z%5E2+%3E=++xy+%2B+xz+%2B+yz\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "we don't even need the non-negativity conditions for x, y, and z. </SPAN>\n" );
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