document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163320>Question 1163320</A>: <I><SPAN STYLE=\"word-break: break-all;\"> let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%28x%29+=+ax%5E3+%2B+6x%5E2+%2B+bx+%2B+4+ ALIGN=MIDDLE  ALT=\"+f%28x%29+=+ax%5E3+%2B+6x%5E2+%2B+bx+%2B+4+\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Determine the constants a and b so that f has a local minimum at x = -1 and a local maximum at x = 2. Show all your work using calculus.\r<BR>\n" );
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document.write( "Thank you. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #787394 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=787394\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "A local maximum or minimum means the slope of the graph is 0; that means the derivative of the function is 0.<br><BR>\n" );
document.write( "The given function is a 3rd degree polynomial; its derivative will be a 2nd degree polynomial.  That is consistent with the given information that the function has one local minimum and one local maximum.<br><BR>\n" );
document.write( "Find the derivative of the function:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=df%2Fdx+=+3ax%5E2%2B12x%2Bb ALIGN=MIDDLE  ALT=\"df%2Fdx+=+3ax%5E2%2B12x%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Use the fact that the derivative is 0 at x=-1 and x=2 to form two equations in the unknowns a and b.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3a%28-1%29%5E2%2B12%28-1%29%2Bb+=+0 ALIGN=MIDDLE  ALT=\"3a%28-1%29%5E2%2B12%28-1%29%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=3a-12%2Bb+=+0 ALIGN=MIDDLE  ALT=\"3a-12%2Bb+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3a%2Bb+=+12 ALIGN=MIDDLE  ALT=\"3a%2Bb+=+12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3a%282%29%5E2%2B12%282%29%2Bb+=+0 ALIGN=MIDDLE  ALT=\"3a%282%29%5E2%2B12%282%29%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=12a%2B24%2Bb=0 ALIGN=MIDDLE  ALT=\"12a%2B24%2Bb=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(2) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12a%2Bb+=+-24 ALIGN=MIDDLE  ALT=\"12a%2Bb+=+-24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Subtracting (1) from (2)...<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9a+=+-36 ALIGN=MIDDLE  ALT=\"9a+=+-36\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+-4 ALIGN=MIDDLE  ALT=\"a+=+-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Then substituting a=-4 in either (1) or (2) gives b=24.<br><BR>\n" );
document.write( "ANSWER: a = -4; b = 24<br><BR>\n" );
document.write( "A graph...:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-4%2C4%2C-20%2C60%2C-4x%5E3%2B6x%5E2%2B24x%2B4%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-4%2C4%2C-20%2C60%2C-4x%5E3%2B6x%5E2%2B24x%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "---------------------------------------------------------<br><BR>\n" );
document.write( "For tutor @ikleyn.....<br><BR>\n" );
document.write( "My goodness! Grow up!<br><BR>\n" );
document.write( "Do you feel slighted because I felt a solution identical to yours but presented differently might be useful to the student?!<br><BR>\n" );
document.write( "Your solutions are NOT always the solutions that are the best possible for the particular student.  The same solution presented differently might satisfy the student's needs better....<br><BR>\n" );
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