document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1106177>Question 1106177</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the values of a and b so that the function f(x)=(1/3)x^3 + ax^2 +bx will have a local extrema at (3,1).  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #721122 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=721122\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "We know that...<br><BR>\n" );
document.write( "(a) f(3) = 1:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F3%29%283%5E3%29%2Ba%283%5E2%29%2Bb%283%29+=+1 ALIGN=MIDDLE  ALT=\"%281%2F3%29%283%5E3%29%2Ba%283%5E2%29%2Bb%283%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%2B9a%2B3b+=+1 ALIGN=MIDDLE  ALT=\"9%2B9a%2B3b+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9a%2B3b+=+-8 ALIGN=MIDDLE  ALT=\"9a%2B3b+=+-8\"   DISABLED_style= \"max-width:90%; height:auto;\" >  (1)<br><BR>\n" );
document.write( "and<br><BR>\n" );
document.write( "(b) f'(3) = 0:<BR>\n" );
document.write( "f'(x) = x^2+2ax+b<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%2B6a%2Bb+=+0 ALIGN=MIDDLE  ALT=\"9%2B6a%2Bb+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6a%2Bb+=+-9 ALIGN=MIDDLE  ALT=\"6a%2Bb+=+-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >  (2)<br><BR>\n" );
document.write( "Solve the pair of equations (1) and (2).<br><BR>\n" );
document.write( "The numbers turn out to be ugly fractions, so solve for both a and b using elimination.<br><BR>\n" );
document.write( "solve for a (eliminate b):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9a%2B3b+=+-8 ALIGN=MIDDLE  ALT=\"9a%2B3b+=+-8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18a%2B3b+=+-27 ALIGN=MIDDLE  ALT=\"18a%2B3b+=+-27\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9a+=+-19 ALIGN=MIDDLE  ALT=\"9a+=+-19\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+-19%2F9 ALIGN=MIDDLE  ALT=\"a+=+-19%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "solve for b (eliminate a):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18a%2B6b+=+-16 ALIGN=MIDDLE  ALT=\"18a%2B6b+=+-16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18a%2B3b+=+-27 ALIGN=MIDDLE  ALT=\"18a%2B3b+=+-27\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3b+=+11 ALIGN=MIDDLE  ALT=\"3b+=+11\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+11%2F3 ALIGN=MIDDLE  ALT=\"b+=+11%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The required function is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%281%2F3%29x%5E3-%2819%2F9%29x%5E2%2B%2811%2F3%29x ALIGN=MIDDLE  ALT=\"f%28x%29+=+%281%2F3%29x%5E3-%2819%2F9%29x%5E2%2B%2811%2F3%29x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The graph shows a relative minimum at (3,1):<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-2%2C8%2C-10%2C10%2C%281%2F3%29x%5E3-%2819%2F9%29x%5E2%2B%2811%2F3%29x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );