document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1205103>Question 1205103</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The revenue function for a particular product is R(x)= x(4-0.0001x).How to Find the largest possible revenue.\r<BR>\n" );
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document.write( "to solve this do I just have to find the critical point and check if its the maximum point using the derivative? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #841741 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=841741\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%28x%29=+x%284-0.0001x%29 ALIGN=MIDDLE  ALT=\"R%28x%29=+x%284-0.0001x%29\"   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=R%28x%29=+4x-0.0001x%5E2 ALIGN=MIDDLE  ALT=\"R%28x%29=+4x-0.0001x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "using calculus:<BR>\n" );
document.write( "first derivate\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28d%2Fdx%29%284x-0.0001x%5E2%29=4-2%2A0.0001x=4+-+0.0002x ALIGN=MIDDLE  ALT=\"%28d%2Fdx%29%284x-0.0001x%5E2%29=4-2%2A0.0001x=4+-+0.0002x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "equal it to zero\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4+-+0.0002x=0 ALIGN=MIDDLE  ALT=\"4+-+0.0002x=0\"   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=4+=0.0002x ALIGN=MIDDLE  ALT=\"4+=0.0002x\"   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=4%2F0.0002 ALIGN=MIDDLE  ALT=\"x=4%2F0.0002\"   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=20000 ALIGN=MIDDLE  ALT=\"x=20000\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "plug it in given equation \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%28x%29=+4%2A%2820000%29-0.0001%2A%2820000%29%5E2=40000 ALIGN=MIDDLE  ALT=\"R%28x%29=+4%2A%2820000%29-0.0001%2A%2820000%29%5E2=40000\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "max revenue is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=40000 ALIGN=MIDDLE  ALT=\"40000\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "or, solve it using algebra:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%28x%29=+4x-0.0001x%5E2 ALIGN=MIDDLE  ALT=\"R%28x%29=+4x-0.0001x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The revenue equation is a quadratic, so its graph is a parabola.  Since the coefficient of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > term is negative (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.0001 ALIGN=MIDDLE  ALT=\"-0.0001\"   DISABLED_style= \"max-width:90%; height:auto;\" >), it's an inverted parabola with the vertex at the top.  <BR>\n" );
document.write( "The vertex will thus be the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=maximum ALIGN=MIDDLE  ALT=\"maximum\"   DISABLED_style= \"max-width:90%; height:auto;\" > revenue.  <BR>\n" );
document.write( "To find the vertex, convert <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the vertex form. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R+=+a%28x-h%29%5E2%2Bk ALIGN=MIDDLE  ALT=\"R+=+a%28x-h%29%5E2%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" > where (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" >) is the location of the vertex.  \r<BR>\n" );
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document.write( "Convert to the vertex form by completing the square:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%28x%29=+4x-0.0001x%5E2 ALIGN=MIDDLE  ALT=\"R%28x%29=+4x-0.0001x%5E2\"   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=R%28x%29=-0.0001%28+4x%2F-0.0001%2Bx%5E2%29 ALIGN=MIDDLE  ALT=\"R%28x%29=-0.0001%28+4x%2F-0.0001%2Bx%5E2%29\"   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=R%28x%29=-0.0001%28+-40000x%2Bx%5E2%29 ALIGN=MIDDLE  ALT=\"R%28x%29=-0.0001%28+-40000x%2Bx%5E2%29\"   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=R%28x%29=-0.0001%28+b%5E2-40000x%2Bx%5E2%29-%28-0.0001b%5E2%29 ALIGN=MIDDLE  ALT=\"R%28x%29=-0.0001%28+b%5E2-40000x%2Bx%5E2%29-%28-0.0001b%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >...<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=40000%2F2=20000 ALIGN=MIDDLE  ALT=\"b=40000%2F2=20000\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "  rearrange the terms in parenthesis\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%28x%29=-0.0001%28+x%5E2-40000x%2B20000%5E2%29-%28-0.0001%2A20000%5E2%29 ALIGN=MIDDLE  ALT=\"R%28x%29=-0.0001%28+x%5E2-40000x%2B20000%5E2%29-%28-0.0001%2A20000%5E2%29\"   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=R%28x%29=-0.0001%28+x-20000%29%5E2-%28-40000%29 ALIGN=MIDDLE  ALT=\"R%28x%29=-0.0001%28+x-20000%29%5E2-%28-40000%29\"   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=R%28x%29=-0.0001%28+x-20000%29%5E2%2B40000 ALIGN=MIDDLE  ALT=\"R%28x%29=-0.0001%28+x-20000%29%5E2%2B40000\"   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=h=20000 ALIGN=MIDDLE  ALT=\"h=20000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=40000 ALIGN=MIDDLE  ALT=\"k=40000\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "vertex is at (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20000 ALIGN=MIDDLE  ALT=\"20000\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=40000 ALIGN=MIDDLE  ALT=\"40000\"   DISABLED_style= \"max-width:90%; height:auto;\" >)=>the maximum revenue is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=40000 ALIGN=MIDDLE  ALT=\"40000\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );