document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1195370>Question 1195370</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose that the cost function for a product is given by C(x)=0.003x^3+5x+11,719. Find the production level​ (that is, the value of​ x) that will produce the minimum average cost per unit _ <BR>\n" );
document.write( "                      C(x).\r<BR>\n" );
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document.write( "The production level that produces the minimum average cost per unit is x=\r<BR>\n" );
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document.write( "(Round to the nearest whole number as​ needed.)<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 #827845 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=827845\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "x = number of units produced<BR>\n" );
document.write( "C(x) = cost function<BR>\n" );
document.write( "C(x) = 0.003x^3+5x+11719\r<BR>\n" );
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document.write( "A(x) = average cost function = cost per unit<BR>\n" );
document.write( "A(x) = (cost function)/(number of units produced)<BR>\n" );
document.write( "A(x) = C(x)/x<BR>\n" );
document.write( "A(x) = (0.003x^3+5x+11719)/x<BR>\n" );
document.write( "A(x) = (0.003x^3)/x+(5x)/x+(11719/x)<BR>\n" );
document.write( "A(x) = 0.003x^2+5+(11719/x)\r<BR>\n" );
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document.write( "Apply the derivative<BR>\n" );
document.write( "A ' (x) = d/dx[ 0.003x^2+5+(11719/x) ]<BR>\n" );
document.write( "A ' (x) = d/dx[ 0.003x^2 ]+d/dx[ 5 ]+d/dx[ 11719/x ]<BR>\n" );
document.write( "A ' (x) = 0.006x + 0 - 11719/(x^2)<BR>\n" );
document.write( "A ' (x) = 0.006x - 11719/(x^2)\r<BR>\n" );
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document.write( "Note: if y = 1/x, then dy/dx = -1/(x^2)\r<BR>\n" );
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document.write( "Set the derivative equal to zero and solve for x.<BR>\n" );
document.write( "A ' (x) = 0<BR>\n" );
document.write( "0.006x - 11719/(x^2) = 0<BR>\n" );
document.write( "0.006x = 11719/(x^2)<BR>\n" );
document.write( "0.006x*(x^2) = 11719<BR>\n" );
document.write( "0.006x^3 = 11719<BR>\n" );
document.write( "x^3 = 11719/0.006<BR>\n" );
document.write( "x^3 = 1,953,166.66666667<BR>\n" );
document.write( "x = cubeRoot(1,953,166.66666667)<BR>\n" );
document.write( "x = (1,953,166.66666667)^(1/3)<BR>\n" );
document.write( "x = 125.000888882569<BR>\n" );
document.write( "This value is approximate.<BR>\n" );
document.write( "Rounding to the nearest whole number gets to x = 125 which is the final answer.\r<BR>\n" );
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document.write( "Let's check a few values around x = 125\r<BR>\n" );
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document.write( "Plug in x = 123<BR>\n" );
document.write( "A(x) = (0.003x^3+5x+11719)/x<BR>\n" );
document.write( "A(123) = (0.003(123)^3+5(123)+11719)/(123)<BR>\n" );
document.write( "A(123) = 145.663422764228<BR>\n" );
document.write( "A(123) = 145.663\r<BR>\n" );
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document.write( "Plug in x = 124<BR>\n" );
document.write( "A(x) = (0.003x^3+5x+11719)/x<BR>\n" );
document.write( "A(124) = (0.003(124)^3+5(124)+11719)/(124)<BR>\n" );
document.write( "A(124) = 145.636064516129<BR>\n" );
document.write( "A(124) = 145.636\r<BR>\n" );
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document.write( "Plug in x = 125<BR>\n" );
document.write( "A(x) = (0.003x^3+5x+11719)/x<BR>\n" );
document.write( "A(125) = (0.003(125)^3+5(125)+11719)/(125)<BR>\n" );
document.write( "A(125) = 145.627\r<BR>\n" );
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document.write( "Plug in x = 126<BR>\n" );
document.write( "A(x) = (0.003x^3+5x+11719)/x<BR>\n" );
document.write( "A(126) = (0.003(126)^3+5(126)+11719)/(126)<BR>\n" );
document.write( "A(126) = 145.635936507937<BR>\n" );
document.write( "A(126) = 145.636\r<BR>\n" );
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document.write( "Plug in x = 127<BR>\n" );
document.write( "A(x) = (0.003x^3+5x+11719)/x<BR>\n" );
document.write( "A(127) = (0.003(127)^3+5(127)+11719)/(127)<BR>\n" );
document.write( "A(127) = 145.662590551181<BR>\n" );
document.write( "A(127) = 145.663\r<BR>\n" );
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document.write( "Those results are approximate with the exception of 145.627 which is exact.\r<BR>\n" );
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document.write( "Here's a table to organize things:<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >x</TD><TD >A(x)</TD></tr><tr><TD >123</TD><TD >145.663</TD></tr><tr><TD >124</TD><TD >145.636</TD></tr><tr><TD >125</TD><TD >145.627</TD></tr><tr><TD >126</TD><TD >145.636</TD></tr><tr><TD >127</TD><TD >145.663</TD></tr>\n" );
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document.write( "Based on this small sample, we can see that x = 125 leads to A(x) being the smallest.<BR>\n" );
document.write( "I'll let you try out other x values. \r<BR>\n" );
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document.write( "This is what the graph looks like. I used GeoGebra to make the graph.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/wUWJ8JC.png\"><BR>\n" );
document.write( "If you aren't too familiar with calculus, then you can use the \"minimize\" function on your calculator to determine the lowest point. That point is approximately (125.0009, 145.627)<BR>\n" );
document.write( "In GeoGebra, the function is called \"min\". \r<BR>\n" );
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document.write( "Another useful graphing tool is Desmos. The nice thing about this app is that you can simply click on the lowest point to have the coordinates show up. <BR>\n" );
document.write( "You may need to click on the curve itself before clicking on the lowest point.<BR>\n" );
document.write( "Here's a link to the interactive graph page.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.desmos.com/calculator/qme1bkefgv\">https://www.desmos.com/calculator/qme1bkefgv</a>\r<BR>\n" );
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document.write( "Answer: <font color=red>x = 125</font><BR>\n" );
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