document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=389682>Question 389682</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Prove or disprove: The rectangular solid of minimum surface area with fixed volume is a cube. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #276214 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=276214\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Without loss of generality, let the fixed volume be 1, and the dimensions of the rectangular solid be x, y, z. Then,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xyz ALIGN=MIDDLE  ALT=\"xyz\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 1, and we want to show whether\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28xy+%2B+yz+%2B+zx%29+%3E=+6 ALIGN=MIDDLE  ALT=\"2%28xy+%2B+yz+%2B+zx%29+%3E=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xy+%2B+yz+%2B+zx+%3E=+3 ALIGN=MIDDLE  ALT=\"xy+%2B+yz+%2B+zx+%3E=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is true (if it is true, then the solid must have a surface area of at least 6, and the solid of least surface area is a cube).\r<BR>\n" );
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document.write( "Applying the AM-GM inequality,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28xy+%2B+yz+%2B+zx%29%2F3+%3E=+root%283%2C+x%5E2y%5E2z%5E2%29 ALIGN=MIDDLE  ALT=\"%28xy+%2B+yz+%2B+zx%29%2F3+%3E=+root%283%2C+x%5E2y%5E2z%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xyz+=+1 ALIGN=MIDDLE  ALT=\"xyz+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace it into the right-hand side and simplify:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28xy+%2B+yz+%2B+zx%29%2F3+%3E=+root%283%2C+1%29+=+1 ALIGN=MIDDLE  ALT=\"%28xy+%2B+yz+%2B+zx%29%2F3+%3E=+root%283%2C+1%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xy+%2B+yz+%2B+zx+%3E=+3 ALIGN=MIDDLE  ALT=\"xy+%2B+yz+%2B+zx+%3E=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" > as desired. The equality of AM-GM occurs when x = y = z, i.e. the rectangular solid is a cube.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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