document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1181070>Question 1181070</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If g(x) is real valued function defined on set of real numbers satisfying following three conditions:<BR>\n" );
document.write( "(a) g'(0) = 0 (derivative of g at zero is equal to zero)<BR>\n" );
document.write( "(b) g'' (-1) > 0 (second derivative of g at -1 is more than zero)<BR>\n" );
document.write( "(c) g''(x) < 0 if x lies in (0, 2) where (0, 2) os open interval<BR>\n" );
document.write( "What will be grpah of g(x)? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #810913 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13216)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=810913\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "You can't tell what the whole graph is going to look like; you can only determine certain features of the graph.<br><BR>\n" );
document.write( "(a) g'(0) = 0 (derivative of g at zero is equal to zero)<BR>\n" );
document.write( "The instantaneous slope of the graph is 0 at x=0, which means the tangent to the graph at x=0 is horizontal.  This might be a maximum or minimum; or it might be a point of inflection.<br><BR>\n" );
document.write( "(b) g'' (-1) > 0 (second derivative of g at -1 is more than zero)<BR>\n" );
document.write( "Second derivative greater than zero means the graph is concave up; so at x=-1 the graph is concave up.<br><BR>\n" );
document.write( "(c) g''(x) < 0 if x lies in (0, 2) where (0, 2) is open interval<BR>\n" );
document.write( "Second derivative less than zero means the graph is concave down; so the graph is concave down on the interval (0,2).<br><BR>\n" );
document.write( "With the graph concave up at x=-1, the tangent to the graph horizontal at x=0, and the graph concave down on the interval (0,2), the simplest polynomial function will be a cubic polynomial with an inflection point at x=0.<br><BR>\n" );
document.write( "The function g(x)=-x^3 satisfies all those conditions:<br><BR>\n" );
document.write( "g'(x) = -3x^2; g'(0)=0<br><BR>\n" );
document.write( "g''(x) = -6x; g''(-1)=6 which is greater than 0; and g''(x) is negative for all positive values of x, including the interval (0,2).<br><BR>\n" );
document.write( "A graph....<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-4%2C4%2C-50%2C50%2C-x%5E3%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-4%2C4%2C-50%2C50%2C-x%5E3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );