document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1041296>Question 1041296</A>: <I><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29+=+-x%5E3+%2B+%2827%2F2%29+x%5E2+-+60x+%2B+100 ALIGN=MIDDLE  ALT=\"P%28x%29+=+-x%5E3+%2B+%2827%2F2%29+x%5E2+-+60x+%2B+100\"   DISABLED_style= \"max-width:90%; height:auto;\" >, x &#8805; 5 is an approximation of the total profit (in thousands of dollars) from the sale of x hundred thousand tires. Find the number of hundred thousands of tires that must be sold to maximize profit. \r<BR>\n" );
document.write( "\n" );
document.write( "I know the answer is 5, but how exactly do I solve this? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #656291 by </B><A HREF=/tutors/aboutme.mpl?userid=josmiceli><B>josmiceli(19441)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josmiceli><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=josmiceli><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=656291\">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=P%28x%29+=+-x%5E3+%2B+%2827%2F2%29+x%5E2+-+60x+%2B+100 ALIGN=MIDDLE  ALT=\"P%28x%29+=+-x%5E3+%2B+%2827%2F2%29+x%5E2+-+60x+%2B+100\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,x &#8805; 5<BR>\n" );
document.write( "----------------------------------------------<BR>\n" );
document.write( "Is this in a calculus course? If so:<BR>\n" );
document.write( "Find the derivative of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+P+ ALIGN=MIDDLE  ALT=\"+P+\"   DISABLED_style= \"max-width:90%; height:auto;\" > with respect to <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( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+P%5B1%5D+=+-3x%5E2+%2B+27x+-+60+ ALIGN=MIDDLE  ALT=\"+P%5B1%5D+=+-3x%5E2+%2B+27x+-+60+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Set the derivative equal to zero<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-3x%5E2+%2B+27x+-+60+=+0+ ALIGN=MIDDLE  ALT=\"+-3x%5E2+%2B+27x+-+60+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-x%5E2+%2B+9x+-+20+=+0+ ALIGN=MIDDLE  ALT=\"+-x%5E2+%2B+9x+-+20+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28+-x+%2B+4+%29%2A%28+x+-+5%29+=+0+ ALIGN=MIDDLE  ALT=\"+%28+-x+%2B+4+%29%2A%28+x+-+5%29+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" > ( by looking at it )<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+4+ ALIGN=MIDDLE  ALT=\"+x+=+4+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+5+ ALIGN=MIDDLE  ALT=\"+x+=+5+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Which one is the maximum?<BR>\n" );
document.write( "I know they say <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+%3E=+5+ ALIGN=MIDDLE  ALT=\"+x+%3E=+5+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, but I still want<BR>\n" );
document.write( "to know if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+5+ ALIGN=MIDDLE  ALT=\"+x+=+5+\"   DISABLED_style= \"max-width:90%; height:auto;\" > is at a maximum<BR>\n" );
document.write( "----------------------------------------<BR>\n" );
document.write( "For that, take the 2nd derivative of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+P+ ALIGN=MIDDLE  ALT=\"+P+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "It will be negative for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5Bmax%5D+ ALIGN=MIDDLE  ALT=\"+x%5Bmax%5D+\"   DISABLED_style= \"max-width:90%; height:auto;\" > and positive for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5Bmin%5D+ ALIGN=MIDDLE  ALT=\"+x%5Bmin%5D+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+P%5B2%5D+=+-6x+%2B+27+ ALIGN=MIDDLE  ALT=\"+P%5B2%5D+=+-6x+%2B+27+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+P%284%29+=+-6%2A4+%2B+27+ ALIGN=MIDDLE  ALT=\"+P%284%29+=+-6%2A4+%2B+27+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+P%284%29+=+-24+%2B+27+ ALIGN=MIDDLE  ALT=\"+P%284%29+=+-24+%2B+27+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+P%284%29+=+3+ ALIGN=MIDDLE  ALT=\"+P%284%29+=+3+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+P%285%29+=+-6%2A5+%2B+25+ ALIGN=MIDDLE  ALT=\"+P%285%29+=+-6%2A5+%2B+25+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+P%285%29+=+-30+%2B+25+ ALIGN=MIDDLE  ALT=\"+P%285%29+=+-30+%2B+25+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+P%285%29+=+-5+ ALIGN=MIDDLE  ALT=\"+P%285%29+=+-5+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Therefore the maximum is at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+5+ ALIGN=MIDDLE  ALT=\"+x+=+5+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "500,000 tires must be sold to maximize profit<BR>\n" );
document.write( "------------------------<BR>\n" );
document.write( "Here's the plot:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+600%2C+400%2C+-2%2C+10%2C+-10%2C+100%2C+-x%5E3+%2B+%2827%2F2%29%2Ax%5E2+-+60x+%2B+100+%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );