document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=734675>Question 734675</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this question: \r<BR>\n" );
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document.write( "Show that the equation has at least one real root.\r<BR>\n" );
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document.write( "a) cosx=x^3\r<BR>\n" );
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document.write( "b)x^(5)-x^(2)+2x+3=0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #449082 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=449082\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a) Consider <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=cos%28x%29-x%5E3 ALIGN=MIDDLE  ALT=\"f%28x%29=cos%28x%29-x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "It is a continuous function.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%280%29=cos%280%29-0%5E3=1%3E0 ALIGN=MIDDLE  ALT=\"f%280%29=cos%280%29-0%5E3=1%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28pi%2F2%29=cos%28pi%2F2%29-%28pi%2F2%29%5E3=0-pi%5E3%2F8=-pi%5E3%2F8%3C0 ALIGN=MIDDLE  ALT=\"f%28pi%2F2%29=cos%28pi%2F2%29-%28pi%2F2%29%5E3=0-pi%5E3%2F8=-pi%5E3%2F8%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The continuous function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is positive for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and negative for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=pi%2F2 ALIGN=MIDDLE  ALT=\"x=pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > so it must have a zero somewhere in the (0,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F2 ALIGN=MIDDLE  ALT=\"pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >) interval (according to the intermediate value theorem).<BR>\n" );
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document.write( "You could do something similar with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=x%5E5-x%5E2%2B2x%2B3%3E0 ALIGN=MIDDLE  ALT=\"g%28x%29=x%5E5-x%5E2%2B2x%2B3%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "You can easily see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%280%29=3%3E0 ALIGN=MIDDLE  ALT=\"g%280%29=3%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28-10%29=-100000%2B100%2B20%2B3%3C0 ALIGN=MIDDLE  ALT=\"g%28-10%29=-100000%2B100%2B20%2B3%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and that is as much as is needed to prove is that there is one real root. </SPAN>\n" );
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