document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1184358>Question 1184358</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Verify that the obtained function is smooth at all points in your domain:<BR>\n" );
document.write( "f(x)={ 2 , if 0≤x≤2.72<BR>\n" );
document.write( "-----{ln(x^2),if 2.72< x < 20                                                      \r<BR>\n" );
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document.write( "-------{ x-14 , if 20≤x≤25 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #814981 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=814981\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "No, your function is not smooth in the domain [0, 25].  In fact, it's not even continuous.\r<BR>\n" );
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document.write( "What makes it continuous is the following definition: \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=system%28matrix%283%2C2%2C+2%2C%22%2C%22+0%3C=x+%3C=e%2C+ln%28x%5E2%29%2C%22%2C%22+e+%3C+x+%3C+e%5E3%2C+x%2B6+-+e%5E3%2C+%22%2C%22+e%5E3+%3C=x+%3C=+25%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "But even with the correct definition it still is not smooth, as the function is not differentiable at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+e ALIGN=MIDDLE  ALT=\"x+=+e\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+e%5E3 ALIGN=MIDDLE  ALT=\"x+=+e%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "At <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+e ALIGN=MIDDLE  ALT=\"x+=+e\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the left-hand derivative is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22f%27%22%5Bl%5D+%28e%29=0 ALIGN=MIDDLE  ALT=\"%22f%27%22%5Bl%5D+%28e%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, while the right-hand derivative is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22f%27%22%5Br%5D+%28e%29=2%2Fe ALIGN=MIDDLE  ALT=\"%22f%27%22%5Br%5D+%28e%29=2%2Fe\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so the derivative at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=e ALIGN=MIDDLE  ALT=\"x=e\"   DISABLED_style= \"max-width:90%; height:auto;\" >does not exist.\r<BR>\n" );
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document.write( "At <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+e%5E3 ALIGN=MIDDLE  ALT=\"x+=+e%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the left-hand derivative is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22f%27%22%5Bl%5D+%28e%5E3%29=2%2Fe%5E3 ALIGN=MIDDLE  ALT=\"%22f%27%22%5Bl%5D+%28e%5E3%29=2%2Fe%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, while the right-hand derivative is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22f%27%22%5Br%5D+%28e%5E3%29+=+0 ALIGN=MIDDLE  ALT=\"%22f%27%22%5Br%5D+%28e%5E3%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so the derivative at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=e%5E3 ALIGN=MIDDLE  ALT=\"x=e%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >does not exist.<BR>\n" );
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