document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1048554>Question 1048554</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given f(1-x)+(1-x)f(x)=5 for all real number x, find the maximum value that is attained by f(x).<BR>\n" );
document.write( "Now, this made me think really deep, but I just can't seem to analyze it. <BR>\n" );
document.write( "Please help, thanks. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #664863 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=664863\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%281-x%29%2B%281-x%29f%28x%29=5 ALIGN=MIDDLE  ALT=\"f%281-x%29%2B%281-x%29f%28x%29=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "====> Replacing x by 1-x, we get  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29%2Bxf%281+-+x%29=5 ALIGN=MIDDLE  ALT=\"f%28x%29%2Bxf%281+-+x%29=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Using Cramer's rule to find f(x), \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=%28+abs%28%28matrix%282%2C2%2C5%2C1%2C5%2Cx%29%29%29%29%2F%28abs%28%28matrix%282%2C2%2C1-x%2C1%2C1%2Cx%29%29%29%29+=+%285x-5%29%2F%28x%281-x%29-1%29+=+%285x-5%29%2F%28-x%5E2%2Bx-1%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "(As a check, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%281-x%29+=+%28-5x%29%2F%28-x%5E2%2Bx-1%29 ALIGN=MIDDLE  ALT=\"f%281-x%29+=+%28-5x%29%2F%28-x%5E2%2Bx-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is the expression also obtained when x is replaced by 1-x in the preceding expression for f(x).)\r<BR>\n" );
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document.write( "===> f'(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285x%5E2-10x%29%2F%28-x%5E2+%2B+x-1%29%5E2 ALIGN=MIDDLE  ALT=\"%285x%5E2-10x%29%2F%28-x%5E2+%2B+x-1%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Setting this derivative equal to 0, we get the critical values x = 0, 2.<BR>\n" );
document.write( "By using the first derivative test, we find a local max at x = 0, and a local minimum at x = 2.\r<BR>\n" );
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document.write( "At x = 0, f(0) = 5.<BR>\n" );
document.write( "At x = 2, f(2) = -5/3.\r<BR>\n" );
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document.write( "Therefore, the absolute max value of f(x) is 5, and the absolute min value of f(x) is -5/3.\r<BR>\n" );
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document.write( "We confirm by graphing f(x):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+300%2C+200%2C+-5%2C+5%2C+-5%2C+5%2C+%285x-5%29%2F%28-x%5E2%2Bx-1%29+%29 ALIGN=MIDDLE  ALT=\"graph%28+300%2C+200%2C+-5%2C+5%2C+-5%2C+5%2C+%285x-5%29%2F%28-x%5E2%2Bx-1%29+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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