document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=996110>Question 996110</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let f (x) = 10x^(8/3)\r<BR>\n" );
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document.write( "Find all extreme values (if any) of f on the interval [-8, 8]. Determine at which numbers in the interval these values occur. NOTE: If there is no maximum value (minimum value), then enter N/A in the corresponding blanks.\r<BR>\n" );
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document.write( "Maximum Value:<BR>\n" );
document.write( "f (?)= ?= f (?)\r<BR>\n" );
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document.write( "Minimum Value:<BR>\n" );
document.write( "f (?) = ? \r<BR>\n" );
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document.write( "Please explain<BR>\n" );
document.write( "Thank you </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #614670 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=614670\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Apply the derivative, then solve f ' (x) = 0<BR>\n" );
document.write( "f(x) = 10x^(8/3) <BR>\n" );
document.write( "f ' (x) = 10*(8/3)*x^(8/3-1) <BR>\n" );
document.write( "f ' (x) = (80/3)*x^(5/3) <BR>\n" );
document.write( "0 = (80/3)*x^(5/3) <BR>\n" );
document.write( "(80/3)*x^(5/3) = 0<BR>\n" );
document.write( "x^(5/3) = 0<BR>\n" );
document.write( "[x^(5/3)]^(3/5) = 0^(3/5)<BR>\n" );
document.write( "x = 0^(3/5)<BR>\n" );
document.write( "x = 0<BR>\n" );
document.write( "The only critical value is x = 0\r<BR>\n" );
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document.write( "Make a sign chart for f ' (x) with the critical value 0 plotted on the number line. Also plot -1 and +1 (one number on each side of the critical value)<BR>\n" );
document.write( "If x = -1, then<BR>\n" );
document.write( "f ' (x) = (80/3)*x^(5/3)<BR>\n" );
document.write( "f ' (-1) = (80/3)*(-1)^(5/3)<BR>\n" );
document.write( "f ' (-1) = -26.667 ... negative result<BR>\n" );
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document.write( "If x = 1, then<BR>\n" );
document.write( "f ' (x) = (80/3)*x^(5/3)<BR>\n" );
document.write( "f ' (1) = (80/3)*(1)^(5/3)<BR>\n" );
document.write( "f ' (1) = 26.667 ... positive result<BR>\n" );
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document.write( "We have f ' (x) changing sign as it passes through x = 0<BR>\n" );
document.write( "f ' (x) changes from negative to positive, so we have a local min when x = 0\r<BR>\n" );
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document.write( "f(x) = 10x^(8/3) <BR>\n" );
document.write( "f(0) = 10*0^(8/3) <BR>\n" );
document.write( "f(0) = 0\r<BR>\n" );
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document.write( "The local min is at the point (0,0)<BR>\n" );
document.write( "There is NO local max since there is only one extrema (from the one critical value) </SPAN>\n" );
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