document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1029384>Question 1029384</A>: <I><SPAN STYLE=\"word-break: break-all;\"> i am thinking of 2 numbers. The difference between them is 12. What is the smallest product (multiplication) I can get with my two numbers? Hint: it is not zero! Create a function you can graph. What part of the graph would we call \"smallest\"? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #644482 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><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=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=644482\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let the first number be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X ALIGN=MIDDLE  ALT=\"X\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The second number will be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12-X ALIGN=MIDDLE  ALT=\"12-X\"   DISABLED_style= \"max-width:90%; height:auto;\" > since the sum of the first and second is 12.<BR>\n" );
document.write( "Define the product of the two numbers,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28X%29=X%2812-X%29 ALIGN=MIDDLE  ALT=\"P%28X%29=X%2812-X%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So to find the extrema, put <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28X%29 ALIGN=MIDDLE  ALT=\"P%28X%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > into vertex form,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28X%29=12X-X%5E2 ALIGN=MIDDLE  ALT=\"P%28X%29=12X-X%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28X%29=-X%5E2%2B12X ALIGN=MIDDLE  ALT=\"P%28X%29=-X%5E2%2B12X\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28X%29=-%28X%5E2-12X%2B36%29%2B36 ALIGN=MIDDLE  ALT=\"P%28X%29=-%28X%5E2-12X%2B36%29%2B36\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28X%29=-%28X-6%29%5E2%2B36 ALIGN=MIDDLE  ALT=\"P%28X%29=-%28X-6%29%5E2%2B36\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So what that tells you is that the maximum product of the two numbers is 36 (occurring when X=6) and the parabola opens downward so it's unbounded in the negative direction.<BR>\n" );
document.write( "So there is no minimum value for this function, only a maximum.<BR>\n" );
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