document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=203087>Question 203087</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the absolute maximum and absolute minimum values of the function below. If an absolute maximum or minimum does not exist, enter NONE.<BR>\n" );
document.write( "f(x) = x^2 + 250/x on the open interval (0,infinity ) \r<BR>\n" );
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document.write( "I know that the absolute max is the answer NONE<BR>\n" );
document.write( "but I can not figure out the absolute min<BR>\n" );
document.write( "can someone help please<BR>\n" );
document.write( "thanks </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #153219 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=153219\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here are the basics to finding the extrema (ie the max/min)\r<BR>\n" );
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document.write( "Step 1) Derive the function f(x) to get f'(x)\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f(x) =\frac{x^2 + 250}{x}\"> this means that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f^{\prime}(x) =\frac{d}{dx}\left(\frac{x^2 + 250}{x}\right)=\frac{x^2-250}{x^2}\"> or simply <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f^{\prime}(x)=\frac{x^2-250}{x^2}\">\r<BR>\n" );
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document.write( "Step 2) Set the derivative function f'(x) equal to zero\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 0=\frac{x^2-250}{x^2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x^2-250=0\"> Multiply both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and rearrange the equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x^2=250\"> Add 250 to both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x=\pm \sqrt{250}\"> Take the square root of both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x=\pm 5\sqrt{10}\"> Simplify the square root\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x \approx \pm 15.8114\"> Evaluate the square root\r<BR>\n" );
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document.write( "Now because the domain is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(0,\infty\right)\">, this means that we must reject <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-15.8114 ALIGN=MIDDLE  ALT=\"x=-15.8114\"   DISABLED_style= \"max-width:90%; height:auto;\" > (as it isn't in the domain)\r<BR>\n" );
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document.write( "So the max/min occurs at the approximate value <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=15.8114 ALIGN=MIDDLE  ALT=\"x=15.8114\"   DISABLED_style= \"max-width:90%; height:auto;\" > (or at the endpoints)\r<BR>\n" );
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document.write( "Now if you graph the function (or evaluate a list of x values), you'll find that there is no max value (so you are correct). So this means that the min occurs when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=15.8114 ALIGN=MIDDLE  ALT=\"x=15.8114\"   DISABLED_style= \"max-width:90%; height:auto;\" > or the min occurs at the endpoints of the interval. Because x=0 is not defined (and infinity is not a number), this means that the min must occur at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=15.8114 ALIGN=MIDDLE  ALT=\"x=15.8114\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Step 3) Find the minimum value of the function f(x)\r<BR>\n" );
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document.write( "From here, simply plug in the x value that generates the min value, which in this case is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=15.8114 ALIGN=MIDDLE  ALT=\"x=15.8114\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f(15.8114) =\frac{15.8114^2 + 250}{15.8114}=31.6228\">\r<BR>\n" );
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document.write( "So the absolute min is approximately f(x)=31.6228 and occurs at x=15.8114\r<BR>\n" );
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document.write( "Note: if you want to keep things exact, then you can plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5%2Asqrt%2810%29 ALIGN=MIDDLE  ALT=\"x=5%2Asqrt%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get the min value: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f(5\sqrt{10}) =\frac{(5\sqrt{10})^2 + 250}{5\sqrt{10}}=10\sqrt{10}\">. This would then mean that the absolute min of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=10%2Asqrt%2810%29 ALIGN=MIDDLE  ALT=\"f%28x%29=10%2Asqrt%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > occurs at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5%2Asqrt%2810%29 ALIGN=MIDDLE  ALT=\"x=5%2Asqrt%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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