document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209092>Question 1209092</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The difference of two positive numbers is 9. What is the minimum sum of their squares?\r<BR>\n" );
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document.write( "Absolute extrema/optimization </SPAN>\n" );
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document.write( "Let m and n be the mystery numbers where m > n. <BR>\n" );
document.write( "Both are positive.<BR>\n" );
document.write( "There's a 9 unit gap between them to indicate m-n = 9 which can be rearranged to m = n+9.\r<BR>\n" );
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document.write( "p = m^2+n^2 = summing their squares<BR>\n" );
document.write( "The goal is to make p as small as possible.<BR>\n" );
document.write( "To do so, we need to make n as small as possible.\r<BR>\n" );
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document.write( "If n = 0, then m = n+9 = 0+9 = 9 and m^2+n^2 = 0^2+9^2 = 81<BR>\n" );
document.write( "But n = 0 isn't allowed since n > 0.<BR>\n" );
document.write( "If n approaches 0 from above, then m^2+n^2 approaches 81 from above. \r<BR>\n" );
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document.write( "Therefore, m^2+n^2 doesn't have a minimum. <BR>\n" );
document.write( "We have asymptotic behavior going on. \r<BR>\n" );
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document.write( "A similar situation is to think of the graph y = 1/x<BR>\n" );
document.write( "As x gets bigger, y approaches 0 but never actually gets there. \r<BR>\n" );
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document.write( "Summary: There is no minimum sum of the squares. <BR>\n" );
document.write( "The sum of squares gets closer to 81 from above, but never actually arrives at this sum itself.<BR>\n" );
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