document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192306>Question 1192306</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A marketing firm estimate that n months after the introduction of a client of new product f(n) thousand households will use it, where \r<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 #849055 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849055\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The prompt is incomplete. To find the maximum number of households that will use the product, you need the actual function f(n). \r<BR>\n" );
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document.write( "**Here's how you would generally approach this type of problem:**\r<BR>\n" );
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document.write( "1. **Find the Derivative:**<BR>\n" );
document.write( "   * Calculate the derivative of the function f(n) with respect to 'n'. This will give you the rate of change of the number of households using the product over time.\r<BR>\n" );
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document.write( "2. **Find Critical Points:**<BR>\n" );
document.write( "   * Set the derivative equal to zero and solve for 'n'. These values of 'n' represent potential points where the maximum or minimum number of households might occur.\r<BR>\n" );
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document.write( "3. **Determine the Maximum:**<BR>\n" );
document.write( "   * Use the second derivative test or analyze the behavior of the derivative around the critical points to determine which point corresponds to the maximum number of households.\r<BR>\n" );
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document.write( "4. **Calculate the Maximum Number of Households:**<BR>\n" );
document.write( "   * Substitute the value of 'n' that corresponds to the maximum into the original function f(n). This will give you the maximum number of households that will use the product.\r<BR>\n" );
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document.write( "**Example:**\r<BR>\n" );
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document.write( "Let's say the function is: \r<BR>\n" );
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document.write( "f(n) = 100n - n² \r<BR>\n" );
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document.write( "* **Find the Derivative:**<BR>\n" );
document.write( "   * f'(n) = 100 - 2n\r<BR>\n" );
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document.write( "* **Find Critical Points:**<BR>\n" );
document.write( "   * 100 - 2n = 0<BR>\n" );
document.write( "   * 2n = 100<BR>\n" );
document.write( "   * n = 50\r<BR>\n" );
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document.write( "* **Determine the Maximum:**<BR>\n" );
document.write( "   * Since the coefficient of the n² term is negative, the parabola opens downwards, indicating a maximum at the critical point.\r<BR>\n" );
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document.write( "* **Calculate the Maximum Number of Households:**<BR>\n" );
document.write( "   * f(50) = 100 * 50 - 50² = 5000 - 2500 = 2500 \r<BR>\n" );
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document.write( "**Therefore, in this example, the maximum number of households that will use the product is 2500 thousand, or 2,500,000 households.**<BR>\n" );
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