document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=716373>Question 716373</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use the method of contrapositive proof to prove the following statement.\r<BR>\n" );
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document.write( "Suppose x approaches R. If (x^3) -x is greater than 0, then x is greater than -1. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #439855 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=439855\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "The contrapositive of a statement has the same truth value as the statement.  The contrapositive of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x^3\ -\ x\ >\ 0\ \right\ x\ >\ -1\"> is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\ \leq\ -1\ \right\ x^3\ -\ x\ \leq\ 0\">\r<BR>\n" );
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document.write( "The interval of interest is then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE -\infty\ <\ x\ \leq\ -1\">\r<BR>\n" );
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document.write( "Take the first derivative of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f(x)\ =\ x^3\ -\ x\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ f'(x)\ =\ 3x^2\ -\ 1\">\r<BR>\n" );
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document.write( "Setting the  first derivative equal to zero you can see readily that there are no critical points on the interval of interest.\r<BR>\n" );
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document.write( "The function is continuous and differentiable on the interval <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE -\infty\ <\ x\ \leq\ -1\">.  Choose a test value in this interval, say -2.\r<BR>\n" );
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document.write( "Evaluate the first derivative at the test value:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ f'(-2)\ =\ 3(-2)^2\ -\ 1\ =\ 11\ >\ 0\">\r<BR>\n" );
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document.write( "Therefore the function is increasing on the interval, and therefore the value of the function at the right hand endpoint of the interval is the maximum value of the function on that interval.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ f(-1)\ =\ (-1)^3\ -\ (-1)\ =\ 0\">\r<BR>\n" );
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document.write( "Since the maximum value of the function on the interval of interest is 0, the value of the function is less than or equal to zero whenever <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\ \leq\ -1\">.  Having proven the contrapositive true, the original statement is true.\r<BR>\n" );
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document.write( "Therefore: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x^3\ -\ x\ >\ 0\ \right\ x\ >\ -1\">. Q.E.D.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "<font face=\"Math1\" size=\"+2\">Egw to Beta kai to Sigma</font><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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