document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1035333>Question 1035333</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find max/min when x belongs to the interval (0,infinite sign)<BR>\n" );
document.write( "For function:\r<BR>\n" );
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document.write( "f (x) = (lnx)^2 - 3lnx  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #650008 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=650008\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f+%28x%29+=+%28lnx%29%5E2+-+3lnx+ ALIGN=MIDDLE  ALT=\"f+%28x%29+=+%28lnx%29%5E2+-+3lnx+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  ==>  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=df%28x%29%2Fdx+=+%282lnx-3%29%2Fx ALIGN=MIDDLE  ALT=\"df%28x%29%2Fdx+=+%282lnx-3%29%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now find the critical points of the function by letting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282lnx-3%29%2Fx+=+0 ALIGN=MIDDLE  ALT=\"%282lnx-3%29%2Fx+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2lnx+-+3=0 ALIGN=MIDDLE  ALT=\"2lnx+-+3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+e%5E%283%2F2%29 ALIGN=MIDDLE  ALT=\"x+=+e%5E%283%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+e%5E1.5 ALIGN=MIDDLE  ALT=\"x+=+e%5E1.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now apply the first derivative test:\r<BR>\n" );
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document.write( "When <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E+e%5E1.5 ALIGN=MIDDLE  ALT=\"x+%3E+e%5E1.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=df%28x%29%2Fdx+%3E+0 ALIGN=MIDDLE  ALT=\"df%28x%29%2Fdx+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ==> f(x) is increasing over there.<BR>\n" );
document.write( "When <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C+e%5E1.5 ALIGN=MIDDLE  ALT=\"x+%3C+e%5E1.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=df%28x%29%2Fdx+%3C+0 ALIGN=MIDDLE  ALT=\"df%28x%29%2Fdx+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ==> f(x) is decreasing over there.\r<BR>\n" );
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document.write( "Thus there is a local minimum at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+e%5E1.5 ALIGN=MIDDLE  ALT=\"x+=+e%5E1.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and since it is the only critical point in an open interval (0, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >), it is also an absolute minimum. </SPAN>\n" );
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