document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=138228>Question 138228</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given -x^2-6x-3, is the vertex a maximum or minimum for the function? What is the value fo that maximum or minimum?\r<BR>\n" );
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document.write( "There is nothing in my book about maximums and minimums, please help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #100934 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=100934\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Algebra solution:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"f%28x%29=ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a parabola that opens upward when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3E0 ALIGN=MIDDLE  ALT=\"a%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or downward when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3C0 ALIGN=MIDDLE  ALT=\"a%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (and is no longer a parabola if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=0 ALIGN=MIDDLE  ALT=\"a=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >).  Your value for a is -1, so the parabola opens downward and the vertex is then a maximum.\r<BR>\n" );
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document.write( "The x-coordinate of the vertex is given by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-b%29%2F2a ALIGN=MIDDLE  ALT=\"%28-b%29%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so for your function:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-%28-6%29%29%2F2%28-1%29=3 ALIGN=MIDDLE  ALT=\"%28-%28-6%29%29%2F2%28-1%29=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  The value of the function at that point is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%283%29=-%283%29%5E2-6%283%29-3=-9-18-3=-30 ALIGN=MIDDLE  ALT=\"f%283%29=-%283%29%5E2-6%283%29-3=-9-18-3=-30\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Calculus solution:<BR>\n" );
document.write( "A continuous function has a local extrema wherever the first derivitive is equal to zero.  It is a maximum if the 2nd derivitive is negative at that point and a minimum if the 2nd derivitive is positive at that point.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=y=-x%5E2-6x-3 ALIGN=MIDDLE  ALT=\"f%28x%29=y=-x%5E2-6x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > => <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx=-2x-6 ALIGN=MIDDLE  ALT=\"dy%2Fdx=-2x-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x-6=0 ALIGN=MIDDLE  ALT=\"-2x-6=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > => local extrema at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3 ALIGN=MIDDLE  ALT=\"x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and the value of the function at that point is found the same way as in the algebra solution: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%283%29=-%283%29%5E2-6%283%29-3=-9-18-3=-30 ALIGN=MIDDLE  ALT=\"f%283%29=-%283%29%5E2-6%283%29-3=-9-18-3=-30\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5E2y%2Fdx%5E2=-2 ALIGN=MIDDLE  ALT=\"d%5E2y%2Fdx%5E2=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > => the 2nd derivitive is everywhere negative, so the local extreme point is a maximum. </SPAN>\n" );
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