document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=999360>Question 999360</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the process for finding max/min for functions?<BR>\n" );
document.write( "I have this function: f(x) = 2x^3 - 3x^2 + 6. Find the max/min values of f on [-1,1]<BR>\n" );
document.write( "I know that I must find all the critical points for and that is when f'=0 and the bounded pts, so, x=-1,1.<BR>\n" );
document.write( "Work:<BR>\n" );
document.write( "f'(x) = 6x^2 - 6x = 0<BR>\n" );
document.write( "f'(x) = 6x(x-1) = 0<BR>\n" );
document.write( "//6x = 0<BR>\n" );
document.write( "//x=0<BR>\n" );
document.write( "f'(x) = x-1 = 0<BR>\n" );
document.write( "f'(x) = x = -1<BR>\n" );
document.write( "Hence, x = -1,1,0,-1<BR>\n" );
document.write( "So, these are all my critical values<BR>\n" );
document.write( "Then I take the second derivative:<BR>\n" );
document.write( "f\"(x) = 12x - 6<BR>\n" );
document.write( "then I plug the critical values into the second derivative, then thing that bothers me is I don't understand is why this produces a relevant result.<BR>\n" );
document.write( "Work:<BR>\n" );
document.write( "f\"(x) = 12x - 6<BR>\n" );
document.write( "f\"(0) = 12(0) - 6 = -6 <0, MIN<BR>\n" );
document.write( "f\"(-1) = 12(-1) - 6 = -18 <0, MIN<BR>\n" );
document.write( "f\"(1) = 12(1) - 6 = +6 >0, MAX\r<BR>\n" );
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document.write( "At this point I hit a wall. Am I suppose to plug these output values of the 2nd derivative back into the original function. Or am I completely missing steps?\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "Also, another confusion I have is do the critical values represent x coordinates and the max and min represent the y coordinate and together they give (x,y)?\r<BR>\n" );
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document.write( "Thank you for any and all help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #617032 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52781)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=617032\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> What is the process for finding max/min for functions?<BR>\n" );
document.write( "I have this function: f(x) = 2x^3 - 3x^2 + 6. Find the max/min values of f on [-1,1]<BR>\n" );
document.write( "I know that I must find all the critical points for and that is when f'=0 and the bounded pts, so, x=-1,1.<BR>\n" );
document.write( "Work:<BR>\n" );
document.write( "f'(x) = 6x^2 - 6x = 0<BR>\n" );
document.write( "f'(x) = 6x(x-1) = 0<BR>\n" );
document.write( "//6x = 0<BR>\n" );
document.write( "//x=0<BR>\n" );
document.write( "f'(x) = x-1 = 0<BR>\n" );
document.write( "f'(x) = x = -1<BR>\n" );
document.write( "Hence, x = -1,1,0,-1<BR>\n" );
document.write( "So, these are all my critical values<BR>\n" );
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document.write( "Hello,\r<BR>\n" );
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document.write( "till this point you acted absolutely right.<BR>\n" );
document.write( "In this way you found local maximums and minimums &nbsp;(or, &nbsp;at least, &nbsp;the points that are candidates for local maximum \r<BR>\n" );
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document.write( "and minimum). \r<BR>\n" );
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document.write( "In addition, &nbsp;you need understand these two points:\r<BR>\n" );
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document.write( "1. &nbsp;In order to make a qualified suggestion on global maximum and minimum, &nbsp;you need to check and to compare &nbsp;<U>also</U>&nbsp; \r<BR>\n" );
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document.write( "the &nbsp;<U>values of the function at the ends of the given segment</U>. &nbsp;The derivatives can be non-zero there; &nbsp;nevertheless, \r<BR>\n" );
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document.write( "the global maximum and/or minimum can be located there. \r<BR>\n" );
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document.write( "2. &nbsp;Zero derivative of the given function do not necessary provides local maximum or minimum. &nbsp;They provide only &nbsp;<U>candidates</U>&nbsp; \r<BR>\n" );
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document.write( "to this role. &nbsp;An example is the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3 ALIGN=MIDDLE  ALT=\"x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > &nbsp;at &nbsp;x = 0. &nbsp;It has the zero derivative at this point, &nbsp;but it is neither maximum \r<BR>\n" );
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document.write( "nor minimum. See the plot below.\r<BR>\n" );
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document.write( "<TABLE >\n" );
document.write( " \r\n" );
document.write( "  <TR>\r\n" );
document.write( "  <TD > \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+330%2C+330%2C+-5.5%2C+5.5%2C+-5.5%2C+5.5%2C%0D%0A++++++++++x%5E3+%0D%0A%29 ALIGN=MIDDLE  ALT=\"graph%28+330%2C+330%2C+-5.5%2C+5.5%2C+-5.5%2C+5.5%2C%0D%0A++++++++++x%5E3+%0D%0A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<B>Figure</B>. Plot y = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3 ALIGN=MIDDLE  ALT=\"x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "  </TD>\r\n" );
document.write( "  </TR>\r\n" );
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document.write( "</TABLE>\r<BR>\n" );
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document.write( "I am continuing.\r<BR>\n" );
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document.write( "----------------------------------------------------------------<BR>\n" );
document.write( "Then I take the second derivative:<BR>\n" );
document.write( "f\"(x) = 12x - 6<BR>\n" );
document.write( "then I plug the critical values into the second derivative, then thing that bothers me is I don't understand is why this produces a relevant result.<BR>\n" );
document.write( "Work:<BR>\n" );
document.write( "f\"(x) = 12x - 6<BR>\n" );
document.write( "f\"(0) = 12(0) - 6 = -6 <0, MIN<BR>\n" );
document.write( "f\"(-1) = 12(-1) - 6 = -18 <0, MIN<BR>\n" );
document.write( "f\"(1) = 12(1) - 6 = +6 >0, MAX <BR>\n" );
document.write( "At this point I hit a wall. Am I suppose to plug these output values of the 2nd derivative back into the original function. Or am I completely missing steps? \r<BR>\n" );
document.write( "\n" );
document.write( "Also, another confusion I have is do the critical values represent x coordinates and the max and min represent the y coordinate and together they give (x,y)? <BR>\n" );
document.write( "Thank you for any and all help!<BR>\n" );
document.write( "----------------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "The second derivatives helps you and serves you to determine if the plot of the function is convex downward or concave upward. \r<BR>\n" );
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document.write( "If the second derivative is positive then the function &nbsp;(or the plot of the function)&nbsp; is convex downward  &nbsp;(like &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >).\r<BR>\n" );
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document.write( "If the second derivative is negative then the function (or the plot of the function) is concave upward &nbsp;(like &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x%5E2 ALIGN=MIDDLE  ALT=\"-x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >).\r<BR>\n" );
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document.write( "So, &nbsp;the second derivative helps you to determine, if your candidate to the local minimum or the local maximum is real local \r<BR>\n" );
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document.write( "minimum or local maximum.\r<BR>\n" );
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document.write( "Is this explanation clear to you?\r<BR>\n" );
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document.write( "Please let me know. You may put your answer in the &nbsp;\"Comments for students\" &nbsp;section. \r<BR>\n" );
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document.write( "If you do, &nbsp;do not forget to put the number of the problem &nbsp;(# 999360) &nbsp;in order I could identify it.\r<BR>\n" );
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document.write( "Thank you.\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );