document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1089124>Question 1089124</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the interval where f(x)=lnx/x (x>0) decreases? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #703455 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=703455\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the derivative of ln(x) / x is equal to (ln(x) - 1) / x^2\r<BR>\n" );
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document.write( "set this equal to 0 and you'll find a max/min point on the equation of y = ln(x) / x.\r<BR>\n" );
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document.write( "start with (ln(x) - 1) / x^2 = 0\r<BR>\n" );
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document.write( "multiply both sides of the equation by x^2 to get ln(x) - 1 = 0\r<BR>\n" );
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document.write( "add 1 to both sides of the equation to get ln(x) = 1\r<BR>\n" );
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document.write( "this is true if and only if e^1 = x\r<BR>\n" );
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document.write( "solve for x to get x = e which is equal to 2.718281828\r<BR>\n" );
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document.write( "that's a max/min point on the graph.\r<BR>\n" );
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document.write( "looking at the graph, you see that it's a max point.\r<BR>\n" );
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document.write( "this means the value of ln(x) / x will be increasing to the left of this max point and decreasing to the right of this max point.\r<BR>\n" );
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document.write( "the interval where ln(x)/x is decreasing is therefore from x = 2.718281828..... to positive infinity.\r<BR>\n" );
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document.write( "in interval notation, i believe this will be [2.718281828..., positive infinity)\r<BR>\n" );
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document.write( "the following graph shows this to be true.\r<BR>\n" );
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document.write( "the graph rounds the solution to 3 decimal places.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2017/072904.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "as x gets larger, the function ln(x) / x will approach 0 but will never reach 0.\r<BR>\n" );
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document.write( "note that i didn't figure out what the derivative was.\r<BR>\n" );
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document.write( "i probably could if i remembered how to do it.\r<BR>\n" );
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document.write( "since i didn't, i cheated and used an online derivative calculator.\r<BR>\n" );
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document.write( "it works very well and can be quite useful for you to see if you calculated the derivative correctly.\r<BR>\n" );
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document.write( "that calculator can be found at <a  rel=nofollow HREF= \"http://www.derivative-calculator.net/\" target = \"_blank\">http://www.derivative-calculator.net/</a>\r<BR>\n" );
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