document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=57283>Question 57283</A>: <I><SPAN STYLE=\"word-break: break-all;\"> This is a calculus question:\r<BR>\n" );
document.write( "\n" );
document.write( "Please Please Help! <BR>\n" );
document.write( "Find all the points of inflection (concavity) <BR>\n" );
document.write( "f(x)= x^4-18x^2+4 <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #38982 by </B><A HREF=/tutors/aboutme.mpl?userid=stanbon><B>stanbon(75887)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=stanbon><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=38982\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> f(x)= x^4-18x^2+4 <BR>\n" );
document.write( "f' = 4x^3-36x<BR>\n" );
document.write( "f''=12x^2-36<BR>\n" );
document.write( "Let f''=0:<BR>\n" );
document.write( "12x^2-36=0<BR>\n" );
document.write( "x^2-3=0<BR>\n" );
document.write( "x=sqrt3 or x=-sqrt3<BR>\n" );
document.write( "These are the candidate values:<BR>\n" );
document.write( "Now, does f'' change sign at <BR>\n" );
document.write( "x=sqrt3 or at x=-sqrt3<BR>\n" );
document.write( "Recall f''=12x^2-36<BR>\n" );
document.write( "As x approches sqrt3 from the left <BR>\n" );
document.write( "f'' is negative because 12x^2 will be<BR>\n" );
document.write( "less than 36.<BR>\n" );
document.write( "As x approches sqrt3 from the right f''<BR>\n" );
document.write( "is positive because 12x^2 will be greater<BR>\n" );
document.write( "than 36.<BR>\n" );
document.write( "So there is a point of inflection at x=sqrt3.<BR>\n" );
document.write( "---------<BR>\n" );
document.write( "The same logic will hold for x=-sqrt3 because<BR>\n" );
document.write( "of the term 12x^2 .<BR>\n" );
document.write( "You have two points of inflection: x=sqrt3 and<BR>\n" );
document.write( "x=-sqrt3<BR>\n" );
document.write( "Cheers,<BR>\n" );
document.write( "Stan H. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );