document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200073>Question 1200073</A>: <I><SPAN STYLE=\"word-break: break-all;\"> By using the Intermediate Value Theorem, show that the equation: nth root of x = 1-x has at least one solution in the open interval (0,1) for any positive integer n>=3. \r<BR>\n" );
document.write( "\n" );
document.write( ">= means greater than or equal to </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #834091 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=834091\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%28n%2Cx%29+=+1-x ALIGN=MIDDLE  ALT=\"root%28n%2Cx%29+=+1-x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%28n%2Cx%29-1%2Bx+=+0 ALIGN=MIDDLE  ALT=\"root%28n%2Cx%29-1%2Bx+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+0 ALIGN=MIDDLE  ALT=\"f%28x%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The left hand side is the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+root%28n%2Cx%29-1%2Bx ALIGN=MIDDLE  ALT=\"f%28x%29+=+root%28n%2Cx%29-1%2Bx\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Plug in x = 0<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+root%28n%2Cx%29-1%2Bx ALIGN=MIDDLE  ALT=\"f%28x%29+=+root%28n%2Cx%29-1%2Bx\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%280%29+=+root%28n%2C0%29-1%2B0 ALIGN=MIDDLE  ALT=\"f%280%29+=+root%28n%2C0%29-1%2B0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%280%29+=+0-1%2B0 ALIGN=MIDDLE  ALT=\"f%280%29+=+0-1%2B0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%280%29+=+-1 ALIGN=MIDDLE  ALT=\"f%280%29+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The result is negative.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Plug in x = 1<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+root%28n%2Cx%29-1%2Bx ALIGN=MIDDLE  ALT=\"f%28x%29+=+root%28n%2Cx%29-1%2Bx\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%281%29+=+root%28n%2C1%29-1%2B1 ALIGN=MIDDLE  ALT=\"f%281%29+=+root%28n%2C1%29-1%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%281%29+=+1-1%2B1 ALIGN=MIDDLE  ALT=\"f%281%29+=+1-1%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%281%29+=+1 ALIGN=MIDDLE  ALT=\"f%281%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The result is positive.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "f(x) changes from negative to positive when going through this interval 0 < x < 1.<BR>\n" );
document.write( "Somewhere in the middle we must have f(x) = 0 occur at least once, which leads to f(x) having at least one root in this interval.<BR>\n" );
document.write( "This is because f(x) is a continuous curve.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Identities used:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%28n%2C0%29+=+0 ALIGN=MIDDLE  ALT=\"root%28n%2C0%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%28n%2C1%29+=+1 ALIGN=MIDDLE  ALT=\"root%28n%2C1%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Those two identities are based on<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%5En+=+0 ALIGN=MIDDLE  ALT=\"0%5En+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > where n > 0<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%5En+=+1 ALIGN=MIDDLE  ALT=\"1%5En+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );