document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=79956>Question 79956</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 40. The cost in millions of dollars for a company to manufacture x thousand automobiles is given by the function C(x) = 4x2-40x+225. Find the number of automobiles that must be produced to minimize cost.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #57337 by </B><A HREF=/tutors/aboutme.mpl?userid=doctor_who><B>doctor_who(15)</B></A><img src=\"/images/stars/stars-06.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=doctor_who><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=57337\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A quadratic equation (like the one you have here) forms a \"U\" or \"n\" shaped curve. For the \"U\" type, there is a minimum value at the bottom of the \"U\" and likewise for the \"n\" shape, there is a maximum value at the top of the \"n\".  The actual shape of the graph (ie a \"U\" or an \"n\" shape) depends on the rest of the numbers in the equation.<BR>\n" );
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document.write( "At the minumum (OR MAXIMUM) value of a function, a tangent drawn across the curve at that point will be flat. Which is another way of saying the gradient of the curve at that point is zero.<BR>\n" );
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document.write( "How to find gradients of curves ?  Take the first derivative  (ie \"differentiate it\")\r<BR>\n" );
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document.write( "so C = 4x^2 - 40x + 225 \r<BR>\n" );
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document.write( "dC/dx = 8x - 40\r<BR>\n" );
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document.write( "At the minimum point, dC/dx will equal zero, so :\r<BR>\n" );
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document.write( "0 = 8x - 40\r<BR>\n" );
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document.write( "So 8x = 40<BR>\n" );
document.write( "Therefore x = 5 [thousand automobiles]   (ANSWER)<BR>\n" );
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document.write( "Incidentally, how do we know that this gives a minimum cost and not a maximum one ?  If you differentiate the equation again you get d2C/dx2 = 8 which is positive. So it is a minimum. </SPAN>\n" );
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