document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=126974>Question 126974</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I got 3 for the absolute minimum for f(x) = x^4 – 4x^3 – 5.  Is that correct?\r<BR>\n" );
document.write( "\n" );
document.write( "Also, if you don’t mind, could you check this one to see if I got the right answer?\r<BR>\n" );
document.write( "\n" );
document.write( "The area below y = 4 – x^2 over the interval [0, 2] is 8/3. Is this correct or not?\r<BR>\n" );
document.write( "\n" );
document.write( "Thank you so much!<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #93012 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=93012\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> (3,f(3)) is at least a local minimum because the first derivitive goes to 0 at 3 and the second derivitive is positive at 3.  The picture proves that the minimum is absolute rather convincingly.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-5%2C5%2C-40%2C40%2Cx%5E4-4x%5E3-5%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C600%2C-5%2C5%2C-40%2C40%2Cx%5E4-4x%5E3-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2nd problem.  \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%284-x%5E2%2Cdx%2C0%2C2%29 ALIGN=MIDDLE  ALT=\"int%284-x%5E2%2Cdx%2C0%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%282%29-%282%5E3%2F3%29%29-%284%280%29-%280%5E3%2F3%29%29=8-8%2F3=16%2F3 ALIGN=MIDDLE  ALT=\"%284%282%29-%282%5E3%2F3%29%29-%284%280%29-%280%5E3%2F3%29%29=8-8%2F3=16%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "See the graph below for an illustration of the unreasonability of your answer.  The secant line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2x%2B4 ALIGN=MIDDLE  ALT=\"y=-2x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > forms a right triangle with the axes with an area of 4.  By inspection you can see that the area of this triangle is less than the desired area under the function between the given limits, yet your answer of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%2F3 ALIGN=MIDDLE  ALT=\"8%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is clearly less than 4.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28600%2C600%2C-5%2C5%2C-5%2C5%2C%0D%0Agrid%281%29%2C%0D%0Agraph%28600%2C600%2C-5%2C5%2C-5%2C5%2C4-x%5E2%2C-2x%2B4%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );