document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1079111>Question 1079111</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For the following functions, determine the stationary points and classify them. include graphs.\r<BR>\n" );
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document.write( "a) f(x) = 2x^2 - x + 6<BR>\n" );
document.write( "b) f(x) = X^2 - 4x + 3<BR>\n" );
document.write( "c) f(x) = x^3 - 3x + 3<BR>\n" );
document.write( "d) f(x) = 1 - 9x - 6x^2 -X^3 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #693489 by </B><A HREF=/tutors/aboutme.mpl?userid=Boreal><B>Boreal(15235)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Boreal><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=Boreal><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=693489\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The first two are the vertex of a parabola, and they are minima.<BR>\n" );
document.write( "The first has a vertex at x=-1/4 and y=51/8<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-10%2C10%2C-10%2C10%2C2x%5E2-x%2B6%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-10%2C10%2C-10%2C10%2C2x%5E2-x%2B6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The vertex is at x=2, y=-1<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-10%2C10%2C-10%2C10%2Cx%5E2-4x%2B3%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-10%2C10%2C-10%2C10%2Cx%5E2-4x%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This has a derivative of 3x^2-3, so critical points are when 3x^2-3=0, which occurs when x=1 or -1<BR>\n" );
document.write( "When x=-1, y=5 and the second derivative is 6x and negative when x=-1, so that is a local maximum.  When x=1, a local minimum, y=1.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-10%2C10%2C-10%2C10%2Cx%5E3-3x%2B3%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-10%2C10%2C-10%2C10%2Cx%5E3-3x%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This has a derivative of -3x^2-12x-9, and setting that equal to 0 and factoring out a -3, x^2+4x+3=0, with stationary points at x=-1,-3 and y=5, 1 or (-1, 5) and (-3, 1).  Second derivative is -6x-12, so when x=-1 second derivative is negative and a local maximum, and when x=-3, it is positive and a local minimum.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-10%2C10%2C-10%2C10%2C1-9x-6x%5E2-x%5E3%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-10%2C10%2C-10%2C10%2C1-9x-6x%5E2-x%5E3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
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