document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=123462>Question 123462</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The profit function for a business is given by the equation P(x)=-4x^2+16x-7, where x is the number sold, in thousands, and P(x) is dollars in thousands. Find the maximum profit and how many items must be sold to achieve it. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #90976 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=90976\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Maximize <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=-4x%5E2%2B16x-7 ALIGN=MIDDLE  ALT=\"P%28x%29=-4x%5E2%2B16x-7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Two ways to solve this.\r<BR>\n" );
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document.write( "Without calculus:\r<BR>\n" );
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document.write( "This function describes a convex down parabola, so the maximum point is at the vertex.  The x-coordinate of the vertex of parabola <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 at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-b%2F2a ALIGN=MIDDLE  ALT=\"x=-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=x%5Bmax%5D=-16%2F2%28-4%29=2 ALIGN=MIDDLE  ALT=\"x%5Bmax%5D=-16%2F2%28-4%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The value of the function at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the maximum profit.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%282%29=-4%282%29%5E2%2B16%282%29-7=-16%2B32-y=9 ALIGN=MIDDLE  ALT=\"f%282%29=-4%282%29%5E2%2B16%282%29-7=-16%2B32-y=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the profit is a maximum $9,000 when 2 units are sold.\r<BR>\n" );
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document.write( "Calculus:<BR>\n" );
document.write( "A local minimum or maximum of a function is at the point where the first derivitive is equal to zero\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx=-8x%2B16 ALIGN=MIDDLE  ALT=\"dy%2Fdx=-8x%2B16\"   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=-8x%2B16=0 ALIGN=MIDDLE  ALT=\"-8x%2B16=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8x=-16 ALIGN=MIDDLE  ALT=\"-8x=-16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If the second derivitive evaluated at this value is < 0, the point is a maximum.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5E2%28y%29%2Fdx%5E2=-8 ALIGN=MIDDLE  ALT=\"d%5E2%28y%29%2Fdx%5E2=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Therefore the point is a maximum.\r<BR>\n" );
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document.write( "Find the function value at the maximum point just like before.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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