document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171448>Question 1171448</A>: <I><SPAN STYLE=\"word-break: break-all;\"> ``The value of function -10000e^(-0.03x)-120x+10000 at its critical points is </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850928 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=850928\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To find the value of the function at its critical points, we need to:\r<BR>\n" );
document.write( "\n" );
document.write( "1.  **Find the derivative of the function.**<BR>\n" );
document.write( "2.  **Set the derivative equal to zero and solve for x.** This will give us the critical points.<BR>\n" );
document.write( "3.  **Plug the critical points back into the original function to find the function's value.**\r<BR>\n" );
document.write( "\n" );
document.write( "Let's do this step-by-step:\r<BR>\n" );
document.write( "\n" );
document.write( "**1. Find the derivative:**\r<BR>\n" );
document.write( "\n" );
document.write( "The given function is:\r<BR>\n" );
document.write( "\n" );
document.write( "f(x) = -10000e^(-0.03x) - 120x + 10000\r<BR>\n" );
document.write( "\n" );
document.write( "The derivative, f'(x), is:\r<BR>\n" );
document.write( "\n" );
document.write( "f'(x) = -10000 * (-0.03) * e^(-0.03x) - 120<BR>\n" );
document.write( "f'(x) = 300e^(-0.03x) - 120\r<BR>\n" );
document.write( "\n" );
document.write( "**2. Set the derivative to zero and solve for x:**\r<BR>\n" );
document.write( "\n" );
document.write( "300e^(-0.03x) - 120 = 0<BR>\n" );
document.write( "300e^(-0.03x) = 120<BR>\n" );
document.write( "e^(-0.03x) = 120 / 300<BR>\n" );
document.write( "e^(-0.03x) = 0.4\r<BR>\n" );
document.write( "\n" );
document.write( "Take the natural logarithm of both sides:\r<BR>\n" );
document.write( "\n" );
document.write( "ln(e^(-0.03x)) = ln(0.4)<BR>\n" );
document.write( "-0.03x = ln(0.4)<BR>\n" );
document.write( "x = ln(0.4) / -0.03<BR>\n" );
document.write( "x ≈ 30.543\r<BR>\n" );
document.write( "\n" );
document.write( "**3. Plug the critical point back into the original function:**\r<BR>\n" );
document.write( "\n" );
document.write( "f(x) = -10000e^(-0.03x) - 120x + 10000<BR>\n" );
document.write( "f(30.543) = -10000e^(-0.03 * 30.543) - 120 * 30.543 + 10000<BR>\n" );
document.write( "f(30.543) = -10000e^(-0.91629) - 3665.16 + 10000<BR>\n" );
document.write( "f(30.543) = -10000 * 0.4 + 6334.84<BR>\n" );
document.write( "f(30.543) = -4000 + 6334.84<BR>\n" );
document.write( "f(30.543) = 2334.84\r<BR>\n" );
document.write( "\n" );
document.write( "**Therefore, the value of the function at its critical point is approximately 2334.84.**<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );