document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=202796>Question 202796</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find the absolute maximum and minimum values on the closed interval [-1,8] for the function below.  <BR>\n" );
document.write( "f(x)=x^(2/3)+5\r<BR>\n" );
document.write( "\n" );
document.write( "not sure how to due<BR>\n" );
document.write( "thanks<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #152965 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=152965\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Unfortunately Algebra.com's graphing software will not graph this functino. So I will have to describe the solution. (If you have access to a graphing calculator (or some other device that will graph function), use it to see what the the graph of this functino look like. It will help make sense of my explanation.)<br><BR>\n" );
document.write( "In general, to find absolute maximum and absolute minimum values you<ol><li>Find the values of the function<ul><li>At each of the endpoints of the interval</li><li>at \"bumps\" and \"pointed-parts\" in between the endpoints</li></ul></li><li>Compare all these values and identify the maximum and minimum values.</li></ol><BR>\n" );
document.write( "Before we start, I am going to rewrite the function without the fractional exponent because it will help us as we proceed:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+x%5E%282%2F3%29+%2B+5+=+%28root%283%2C+x%29%29%5E2+%2B+5 ALIGN=MIDDLE  ALT=\"f%28x%29+=+x%5E%282%2F3%29+%2B+5+=+%28root%283%2C+x%29%29%5E2+%2B+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we'll find the values of the functions at the endpoints:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-1%29+=+%28root%283%2C+-1%29%29%5E2+%2B+5 ALIGN=MIDDLE  ALT=\"f%28-1%29+=+%28root%283%2C+-1%29%29%5E2+%2B+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C+-1%29+=+-1 ALIGN=MIDDLE  ALT=\"root%283%2C+-1%29+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-1%29+=+%28-1%29%5E2+%2B+5+=+1+%2B+5+=+6 ALIGN=MIDDLE  ALT=\"f%28-1%29+=+%28-1%29%5E2+%2B+5+=+1+%2B+5+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%288%29+=+%28root%283%2C+8%29%29%5E2+%2B+5 ALIGN=MIDDLE  ALT=\"f%288%29+=+%28root%283%2C+8%29%29%5E2+%2B+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C+8%29+=+2 ALIGN=MIDDLE  ALT=\"root%283%2C+8%29+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%288%29+=+%282%29%5E2+%2B+5+=+4+%2B+5+=+9 ALIGN=MIDDLE  ALT=\"f%288%29+=+%282%29%5E2+%2B+5+=+4+%2B+5+=+9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "To find \"bumps\" and \"pointed-parts\" of a graph one would usually use Calculus and find where the first derivative is zero (a \"bump\") or where it does not exist (a \"pointed-part\"). The first derivative of f(x) is: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F%283root%283%2C+x%29%29 ALIGN=MIDDLE  ALT=\"2%2F%283root%283%2C+x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Since the numerator is 2 (and cannot become 0), the fraction as a while can never be 0. (Think about it.) On the other hand, if x = 0 then the denominator would end up being 0. But we cannot let this happen. This means the first derivative does not exist when x = 0. (In other words, the graph has a \"pointed-part\" at x = 0.) So this is one of the points we should check when we are looking for absolute maximum and minimum values of a function.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%280%29+=+%28root%283%2C+0%29%29%5E2+%2B+5+=+0+%2B+5+=+5 ALIGN=MIDDLE  ALT=\"f%280%29+=+%28root%283%2C+0%29%29%5E2+%2B+5+=+0+%2B+5+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So the absolute maxmum and minimum values must come from these three:<BR>\n" );
document.write( "f(-1) = 6<BR>\n" );
document.write( "f(0) = 5<BR>\n" );
document.write( "f(8) = 9<BR>\n" );
document.write( "So the absolute maximum value is 9 (when x = 8) and the absolute minimum value is 5 (when x = 0). </SPAN>\n" );
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