document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1138898>Question 1138898</A>: <I><SPAN STYLE=\"word-break: break-all;\"> sketch a graph of a function f(x) on the interval -2 (less than or equal to) x (less than or equal to) 2 with the following properties. Label any points on the x-axis or y-axis which are important.<BR>\n" );
document.write( "f is odd.<BR>\n" );
document.write( "f\"(x) is positive for all x>0.<BR>\n" );
document.write( "f'(0) is negative.<BR>\n" );
document.write( "f'(1) = 0.<BR>\n" );
document.write( "f(2) = 0.\r<BR>\n" );
document.write( "\n" );
document.write( "thanks for helping!! </SPAN>\n" );
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document.write( "(1) f(x) is odd.<br><BR>\n" );
document.write( "That means the graph is symmetric with respect to the origin.<BR>\n" );
document.write( "That in turn means that f(0) = 0.<br><BR>\n" );
document.write( "(2) f''(x) is positive for all x > 0.<br><BR>\n" );
document.write( "That means the graph is concave up for all x > 0.<BR>\n" );
document.write( "The fact that f(x) is odd then means the graph is concave down for all x < 0.<br><BR>\n" );
document.write( "(3) f'(0) is negative.<br><BR>\n" );
document.write( "That means the slope of the graph at x=0 is negative (downhill to the right).<br><BR>\n" );
document.write( "(4) f'(1) = 0.<br><BR>\n" );
document.write( "That means the slope of the graph is 0 (the tangent to the graph is horizontal) at x=1.<BR>\n" );
document.write( "That in turn means the slope is also 0 at x = -1.<br><BR>\n" );
document.write( "(5) f(2) = 0.<br><BR>\n" );
document.write( "That means that f(-2) = 0 also.<br><BR>\n" );
document.write( "I can't draw a graph of such a function, because no kind of graph that I know how to graph with the tools on this forum has all those characteristics.<br><BR>\n" );
document.write( "The graph below of the polynomial <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x%5E3-4x+=%28x%2B2%29%28x%29%28x-2%29 ALIGN=MIDDLE  ALT=\"y+=+x%5E3-4x+=%28x%2B2%29%28x%29%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > almost works, except that the slopes of 0 are not at exactly x=1 and x=-1.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-3%2C3%2C-10%2C10%2Cx%5E3-4x%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-3%2C3%2C-10%2C10%2Cx%5E3-4x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );