document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1117429>Question 1117429</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the coordinates of the inflection points for f(x)=xe^x?<BR>\n" );
document.write( "Calculus   </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #732507 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=732507\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "Possible inflection points exist where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{d^2}{dx^2}f(x)\ =\ 0\">\r<BR>\n" );
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document.write( "Such a point is a point of inflection if the concavity of the function changes from one side of the critical point to the other.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ f(x)\ =\ xe^x\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{d}{dx}\ =\ xe^x\ =\ (x\,+\,1)e^x\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{d^2}{dx^2}f(x)\ = \frac{d}{dx}(x\,+\,1)e^x\ =\ (x\,+\,2)e^x\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (x\,+\,2)e^x\ =\ 0\">\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^x\ >\ 0\ \forall\ x\ \in\ \mathbb{R}\">, the only point where the second derivative is zero is at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\ =\ -2\">, and therefore the only possible inflection point is at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \(-2,\, -2e^{-2}\)\">\r<BR>\n" );
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document.write( "To test concavity at a point, evaluate the second derivative at the point in question.  If <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{d^2}{dx^2}f(x)\ >\ 0\">, the function is concave downward at that point.  If <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{d^2}{dx^2}f(x)\ >\ 0\">, the function is concave upward.\r<BR>\n" );
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document.write( "Since our critical point is at -2, we can choose -3 and -1 as points on either side to test\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{d^2}{dx^2}f(-3)\ =\ (-3\,+\,2)e^{-3}\ <\ 0\">\r<BR>\n" );
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document.write( "Since we have already determined that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^x\"> is positive on the entire domain, the fact that the lead coefficient is negative makes the value of the second derivative at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\ =\ -3\"> negative.  Consequently, to the left of the critical point, the original function is concave downward.\r<BR>\n" );
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document.write( "Similarly, and I will leave the details to you, evaluating the second derivative at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\ =\ -1\"> results in a finding that the original function is concave upward to the right of the critical point.\r<BR>\n" );
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document.write( "Therefore, by the definition of inflection point, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \(-2,\, -2e^{-2}\)\"> is indeed a point of inflection, and since there was only one root to the equation <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{d^2}{dx^2}f(x)\ =\ 0\">, we are assured that this is the only point of inflection.\r<BR>\n" );
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document.write( "An examination of the graph of the second derivative of the original function superimposed on a graph of the original function should give you the warm fuzzy feeling that the answer derived above is correct.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/display-illustration.mpl?tutor=solver91311&illustration_name=xe_xPtOfInflectionCrop.jpg\">\r<BR>\n" );
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document.write( "A little closer look:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/display-illustration.mpl?tutor=solver91311&illustration_name=xe_xPtOfInflectionZoomCrop.jpg\">\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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