document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=132033>Question 132033</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The sum of two numbers is 18. If the sum of their squares is a minimum find the numbers. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #96502 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=96502\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By=18 ALIGN=MIDDLE  ALT=\"x%2By=18\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or put another way, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=18-x ALIGN=MIDDLE  ALT=\"y=18-x\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "The sum of the two numbers squares is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2By%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2By%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, but because of the first equation, can be expressed as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B%2818-x%29%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2B%2818-x%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is a function in x.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E2%2B%2818-x%29%5E2 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E2%2B%2818-x%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Expand the binomial:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E2%2B324-36x%2Bx%5E2 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E2%2B324-36x%2Bx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=2x%5E2-36x%2B324 ALIGN=MIDDLE  ALT=\"f%28x%29=2x%5E2-36x%2B324\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Two ways to find a minimum for this function.  The algebra way is to recognize that this is a convex up parabola and solve for the vertex.  The calculus way is to set the first derivitive equal to zero and solve.\r<BR>\n" );
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document.write( "The x-coordinate of the vertex of a parabola in the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"f%28x%29=ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > is found by evaluating <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2F2a ALIGN=MIDDLE  ALT=\"-b%2F2a\"   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=36%2F4=9 ALIGN=MIDDLE  ALT=\"36%2F4=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so the sum of squares is a minimum when x = 9 => y = 9.\r<BR>\n" );
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document.write( "Calculus method:<BR>\n" );
document.write( "Take the first derivitive <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28df%28x%29%29%2Fdx=4x-36 ALIGN=MIDDLE  ALT=\"%28df%28x%29%29%2Fdx=4x-36\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Set it equal to zero and solve<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x-36=0 ALIGN=MIDDLE  ALT=\"4x-36=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x=36 ALIGN=MIDDLE  ALT=\"4x=36\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=9 ALIGN=MIDDLE  ALT=\"x=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so f has a local extreme point at x = 9\r<BR>\n" );
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document.write( "Now take the second derivitive\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28d%5E2f%28x%29%29%2Fdx%5E2=4 ALIGN=MIDDLE  ALT=\"%28d%5E2f%28x%29%29%2Fdx%5E2=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is positive, therefore the local extreme is a minimum.  So the function is minimum when x = 9 => y = 9.\r<BR>\n" );
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document.write( "Same answer. </SPAN>\n" );
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