document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=999407>Question 999407</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Trying to find the intervals where this function increases and decreases but running into some problems.\r<BR>\n" );
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document.write( "So I know to find the critical values first:<BR>\n" );
document.write( "f(x) = x^4 + 2x^3 +1 <BR>\n" );
document.write( "f'(x) = 4x^3 +6x^2<BR>\n" );
document.write( "f'(x) = 2x^2(2x+3)<BR>\n" );
document.write( "x = 0 and x = -3/2 are the critical values.\r<BR>\n" );
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document.write( "Now, if I understand correctly I use the first derivative test to determine in between these two critical values where it's increasing/decreasing.\r<BR>\n" );
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document.write( "Therefore, I make a number line and I plug in values around them into the derivative.\r<BR>\n" );
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document.write( "<----(-2)----(-3/2)----(-1)----(0)----(1)---->\r<BR>\n" );
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document.write( "f'(-2) = 4(-2)^3 + 6(-2)^2 = -8 therefore (-) on the left of (-3/2)<BR>\n" );
document.write( "f'(-1) = 4(-1)^3 + 6(-1)^2 = 2 therefore (+) on the right side (-3/2) and left of (0)<BR>\n" );
document.write( "f'(1) = 4(1)^3 + 6(1) = 10 therefore (+) on the right of (0)\r<BR>\n" );
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document.write( "bit confused on why on the left of (0) there would be another (+) I thought it worked by -,+,-,+ what do I get when I have two (+)'s or two (-)'s in a row like this?<BR>\n" );
document.write( "All I can deduce now is that I have a minimum between -2 and -3/2.\r<BR>\n" );
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document.write( "Maybe I am getting things confused here with other methods.<BR>\n" );
document.write( "But since I have proven I have a (-) the left of (-3/2) does that imply that the function is decreasing from (-inf,-3/2) then increasing from (-3/2, +inf) and thus increasing (0,inf)\r<BR>\n" );
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document.write( "Feel like I am really misunderstanding something here\r<BR>\n" );
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document.write( "Please clarify<BR>\n" );
document.write( "Thank yu\r<BR>\n" );
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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 #617073 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=617073\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> It is possible to have the increasing/decreasing intervals not alternate. There is no rule that says they have to alternate. You can easily have \"decreasing, increasing, increasing\" which is the case here\r<BR>\n" );
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document.write( "Answer: \r<BR>\n" );
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document.write( "Increasing Interval: <IMG STYLE=\"max-width:80%;\" src=\"http://www.sciweavers.org/tex2img.php?eq=%5Cleft%28-%5Cfrac%7B3%7D%7B2%7D%2C%200%5Cright%29%20%5Ccup%20%280%2C%5Cinfty%29&bc=White&fc=Black&im=jpg&fs=12&ff=arev&edit=0\" align=\"center\" border=\"0\" alt=\"\left(-\frac{3}{2}, 0\right) \cup (0,\infty)\" width=\"144\" height=\"47\" />\r<BR>\n" );
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document.write( "Decreasing Interval: <IMG STYLE=\"max-width:80%;\" src=\"http://www.sciweavers.org/tex2img.php?eq=%5Cleft%28-%5Cinfty%2C%20-%5Cfrac%7B3%7D%7B2%7D%5Cright%29&bc=White&fc=Black&im=jpg&fs=12&ff=arev&edit=0\" align=\"center\" border=\"0\" alt=\"\left(-\infty, -\frac{3}{2}\right)\" width=\"94\" height=\"47\" />\r<BR>\n" );
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document.write( "Side Note: the local min occurs at x = -3/2. There is NO local min at x = 0 because we do NOT have a sign change in f ' (x) when we pass through x = 0. </SPAN>\n" );
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