document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=384359>Question 384359</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In the function f(x) = ax 2 + bx + c, the minimum or maximum value occurs where x is equal to -b/(2a).   How is that value derived?  \r<BR>\n" );
document.write( "\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 #272085 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=272085\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Good question. However, the proof requires a little introductory calculus.\r<BR>\n" );
document.write( "\n" );
document.write( "In calculus, we define a function called the \"derivative,\" which measures the instantaneous rate of change of a function. Basically, it is the same as finding slope, except that the two points get infinitely close, and it can be evaluated using limits. The derivative is usually denoted f'(x) or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28d%2Fdx%29+f%28x%29 ALIGN=MIDDLE  ALT=\"%28d%2Fdx%29+f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "\n" );
document.write( "By the power rule (you'll learn it early in calculus), the derivative of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2+%2B+bx+%2B+c ALIGN=MIDDLE  ALT=\"ax%5E2+%2B+bx+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2ax+%2B+b ALIGN=MIDDLE  ALT=\"2ax+%2B+b\"   DISABLED_style= \"max-width:90%; height:auto;\" >. The relative minimum or maximum occurs when the derivative is equal to zero, and the slope is positive on one side of that point and negative on the other.\r<BR>\n" );
document.write( "\n" );
document.write( "Since the derivative of the quadratic is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2ax+%2B+b ALIGN=MIDDLE  ALT=\"2ax+%2B+b\"   DISABLED_style= \"max-width:90%; height:auto;\" > it is easy to see that this derivative is equal to zero when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-b%2F2a ALIGN=MIDDLE  ALT=\"x+=+-b%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" >, therefore it is the vertex. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );