document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1036323>Question 1036323</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the value of a for which <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%28x%29=-2x%5E3%2B3%28a%2B2%29x%5E2-12ax%2B4a%5E2+ ALIGN=MIDDLE  ALT=\"+f%28x%29=-2x%5E3%2B3%28a%2B2%29x%5E2-12ax%2B4a%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > is tangent to the positive x-axis and has a relative maximumn at that point of contact.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #650995 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=650995\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let (b,0) be the point of tangency.\r<BR>\n" );
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document.write( "Now the derivative is given by f'(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6x%5E2+%2B+6%28a%2B2%29x-12a ALIGN=MIDDLE  ALT=\"-6x%5E2+%2B+6%28a%2B2%29x-12a\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Setting this to 0 to find the critical values, we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-%28a%2B2%29x+%2B+2a+=+%28x-2%29%28x-a%29+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+-%28a%2B2%29x+%2B+2a+=+%28x-2%29%28x-a%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "==> x = 2 or x = a.\r<BR>\n" );
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document.write( "Case(i).  Let b = a.<BR>\n" );
document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2a%5E3%2B3%28a%2B2%29a%5E2-12a%5E2%2B4a%5E2+=+0 ALIGN=MIDDLE  ALT=\"-2a%5E3%2B3%28a%2B2%29a%5E2-12a%5E2%2B4a%5E2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%28a-2%29+=+0 ALIGN=MIDDLE  ALT=\"a%5E2%28a-2%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, after reduction.<BR>\n" );
document.write( "==> a = 0 (double root), or a = 2.<BR>\n" );
document.write( "Discard a = 0, because if it is to be the x-coordinate of the point of tangency, a has to be positive.  (Remember tangency to positive x-axis.)<BR>\n" );
document.write( "==> a = 2.<BR>\n" );
document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%28x%29=-2x%5E3%2B12x%5E2-24x%2B16+=+-2%28x-2%29%5E3+ ALIGN=MIDDLE  ALT=\"+f%28x%29=-2x%5E3%2B12x%5E2-24x%2B16+=+-2%28x-2%29%5E3+\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "But this function has to be discarded as well, because even though f'(2) = 0, f\"(2) = 0, and hence there is no maximum at that point, but a point of inflection.\r<BR>\n" );
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document.write( "Case (ii). Let b = 2.<BR>\n" );
document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28b=2%29+=+-16%2B3%28a%2B2%29%2A4-24a%2B4a%5E2+=+0 ALIGN=MIDDLE  ALT=\"f%28b=2%29+=+-16%2B3%28a%2B2%29%2A4-24a%2B4a%5E2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4a%5E2-12a%2B8+=+0 ALIGN=MIDDLE  ALT=\"4a%5E2-12a%2B8+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2+-+3a+%2B+2+=+0 ALIGN=MIDDLE  ALT=\"a%5E2+-+3a+%2B+2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ==> (a-1)(a-2) = 0 ==> a = 1 or a = 2.\r<BR>\n" );
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document.write( "Now we already know what happens when a = 2, and so we proceed letting a = 1.<BR>\n" );
document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%28x%29=-2x%5E3%2B9x%5E2-12x%2B4+=+%28x-2%29%5E2%28-2x%2B1%29+ ALIGN=MIDDLE  ALT=\"+f%28x%29=-2x%5E3%2B9x%5E2-12x%2B4+=+%28x-2%29%5E2%28-2x%2B1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "By using the 2nd derivative test, we find that there is a relative maximum<BR>\n" );
document.write( "at x = 2. (There is relative min at x = 1.)\r<BR>\n" );
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document.write( "Therefore <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28a+=+1%29 ALIGN=MIDDLE  ALT=\"highlight%28a+=+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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