document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=475152>Question 475152</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If the lengths of the sides of a right triangle are, in increasing order, a, b, and c. Prove that a^3+b^3<c^3 ( a cube plus b cube is less than c cube) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #325916 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=325916\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The inequality is true iff\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (a^3 + b^3)^2 < (c^3)^2 = c^6\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (a^3 + b^3)^2 < (a^2 + b^2)^3\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a^6 + 2a^3b^3 + b^6 < a^6 + 3a^4b^2 + 3a^2b^4 + b^6\">\r<BR>\n" );
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document.write( "The a^6 + b^6 on both side cancel out, so we want to show for positive a,b\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2a^3b^3 < 3a^4b^2 + 3a^2b^4\">\r<BR>\n" );
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document.write( "Divide both sides by (a^2)(b^2)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2ab < 3a^2 + 3b^2\">\r<BR>\n" );
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document.write( "By AM-GM inequality, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{3a^2 + 3b^2}{2} \ge \sqrt{9a^2b^2} = 3ab\">\r<BR>\n" );
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document.write( "This implies that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 3a^2 + 3b^2 \ge 6ab > 2ab\">, so we're done. </SPAN>\n" );
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