document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1108445>Question 1108445</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Graph the function -x^3+5x^2-2x+1\r<BR>\n" );
document.write( "\n" );
document.write( "A_Find the value of x at which the given  f(x) has a local maximum. Round to the nearest hundredth.<BR>\n" );
document.write( "f' = -3x^2 + 10x - 2 = 0<BR>\n" );
document.write( "*[invoke solve_quadratic_equation -3,10,-2]<BR>\n" );
document.write( "=================<BR>\n" );
document.write( "x = \r<BR>\n" );
document.write( "\n" );
document.write( "B_Find the value of x at which the given f(x) has a local minimum. Round to the nearest hundredth. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #723472 by </B><A HREF=/tutors/aboutme.mpl?userid=Alan3354><B>Alan3354(69443)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Alan3354><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=Alan3354><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=723472\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Graph the function -x^3+5x^2-2x+1\r<BR>\n" );
document.write( "\n" );
document.write( "A_Find the value of x at which the given  f(x) has a local maximum. Round to the nearest hundredth.<BR>\n" );
document.write( "B_Find the value of x at which the given f(x) has a local minimum. Round to the nearest hundredth.<BR>\n" );
document.write( "=================\r<BR>\n" );
document.write( "\n" );
document.write( "f' = -3x^2 + 10x - 2 = 0<BR>\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=solve_quadratic_equation.solver>SOLVE quadratic equation (work shown, graph etc)</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3x%5E2%2B10x%2B-2+=+0 ALIGN=MIDDLE  ALT=\"-3x%5E2%2B10x%2B-2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%2810%29%5E2-4%2A-3%2A-2=76 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%2810%29%5E2-4%2A-3%2A-2=76\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=76 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28-10%2B-sqrt%28+76+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-10%2B-sqrt%28+76+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D+=+%28-%2810%29%2Bsqrt%28+76+%29%29%2F2%5C-3+=+0.213700352153109 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%2810%29%2Bsqrt%28+76+%29%29%2F2%5C-3+=+0.213700352153109\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%2810%29-sqrt%28+76+%29%29%2F2%5C-3+=+3.11963298118022 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%2810%29-sqrt%28+76+%29%29%2F2%5C-3+=+3.11963298118022\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3x%5E2%2B10x%2B-2 ALIGN=MIDDLE  ALT=\"-3x%5E2%2B10x%2B-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3x%5E2%2B10x%2B-2+=+%28x-0.213700352153109%29%2A%28x-3.11963298118022%29 ALIGN=MIDDLE  ALT=\"-3x%5E2%2B10x%2B-2+=+%28x-0.213700352153109%29%2A%28x-3.11963298118022%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 0.213700352153109, 3.11963298118022.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-3%2Ax%5E2%2B10%2Ax%2B-2+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-3%2Ax%5E2%2B10%2Ax%2B-2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "\n" );
document.write( "=================<BR>\n" );
document.write( "x = 0.21, 3.12<BR>\n" );
document.write( "---<BR>\n" );
document.write( "f\" = -6x + 10<BR>\n" );
document.write( "f\"(0.21) = positive --> minimum<BR>\n" );
document.write( "f\"(3.12) = negative --> maximum<BR>\n" );
document.write( "================================================<BR>\n" );
document.write( "DL the FREE graph software at<BR>\n" );
document.write( "www.padowan.dk<BR>\n" );
document.write( "Use Ins, enter -x^3+5x^2-2x+1\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );