document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1047332>Question 1047332</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please find the minimum and maximum values of the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%5E8%28x%29%2Bcos%5E14%28x%29 ALIGN=MIDDLE  ALT=\"sin%5E8%28x%29%2Bcos%5E14%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #663015 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=663015\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Well, I am sending this commentary to \"ikleyn\"--\r<BR>\n" );
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document.write( "Of course, it will be \"SLIGHTLY SHIFTED\".  \r<BR>\n" );
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document.write( "One MISTAKE you made was that you plotted <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+w%5E7+%2B+%281-w%29%5E4 ALIGN=MIDDLE  ALT=\"z+=+w%5E7+%2B+%281-w%29%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > as if x was the independent variable in the function, and over the SAME COORDINATE SYSTEM!  It was NOT.  \r<BR>\n" );
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document.write( "Remember, I let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+cos%5E2%28x%29 ALIGN=MIDDLE  ALT=\"w+=+cos%5E2%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  <BR>\n" );
document.write( "So naturally, the lowest points of the two graphs will NOT jive.  BUT, what is  guaranteed is that they will have the SAME LOWEST level.\r<BR>\n" );
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document.write( "Another MISTAKE you made was that, z is a composite function, and as such, its domain, as a function of w, is only <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C=+w+%3C=+1 ALIGN=MIDDLE  ALT=\"0+%3C=+w+%3C=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Meaning, your green-colored graph is WRONG.  <BR>\n" );
document.write( "The range of z is [<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28+%28-root%283%2C4%29+%2B+sqrt%28+root%283%2C16%29%2B4root%283%2C28%29%29%29%2F%282%2Aroot%283%2C7%29%29+%29%5E7+%2B+%281-+%28-root%283%2C4%29+%2B+sqrt%28+root%283%2C16%29%2B4root%283%2C28%29%29%29%2F%282%2Aroot%283%2C7%29%29+%29%5E4 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >,1], or[0.05310578625,1] (approximately).\r<BR>\n" );
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document.write( "In fact, I can even predict/estimate the amount of \"shift\" that you're talking about:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E-1%28sqrt%280.5860760201%29%29+-+0.5860760201+=+0.1128151882 ALIGN=MIDDLE  ALT=\"cos%5E-1%28sqrt%280.5860760201%29%29+-+0.5860760201+=+0.1128151882\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Naughty, naughty, not so smarty....\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%5E8%28x%29%2Bcos%5E14%28x%29 ALIGN=MIDDLE  ALT=\"sin%5E8%28x%29%2Bcos%5E14%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%5E2%28x%29%29%5E4%2B%28cos%5E2%28x%29%29%5E7 ALIGN=MIDDLE  ALT=\"%28sin%5E2%28x%29%29%5E4%2B%28cos%5E2%28x%29%29%5E7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281-cos%5E2%28x%29%29%5E4%2B%28cos%5E2%28x%29%29%5E7 ALIGN=MIDDLE  ALT=\"%281-cos%5E2%28x%29%29%5E4%2B%28cos%5E2%28x%29%29%5E7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+cos%5E2%28x%29 ALIGN=MIDDLE  ALT=\"w+=+cos%5E2%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  ===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+0%3C=+w+%3C=+1 ALIGN=MIDDLE  ALT=\"+0%3C=+w+%3C=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and \r<BR>\n" );
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document.write( "===>  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281-cos%5E2%28x%29%29%5E4%2B%28cos%5E2%28x%29%29%5E7+=+w%5E7+%2B+%281-w%29%5E4 ALIGN=MIDDLE  ALT=\"%281-cos%5E2%28x%29%29%5E4%2B%28cos%5E2%28x%29%29%5E7+=+w%5E7+%2B+%281-w%29%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+w%5E7+%2B+%281-w%29%5E4 ALIGN=MIDDLE  ALT=\"z+=+w%5E7+%2B+%281-w%29%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dz%2Fdw+=+7w%5E6+-+4%281-w%29%5E3 ALIGN=MIDDLE  ALT=\"dz%2Fdw+=+7w%5E6+-+4%281-w%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Letting this equal to zero, we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7w%5E6+=+4%281-w%29%5E3 ALIGN=MIDDLE  ALT=\"7w%5E6+=+4%281-w%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C+7%29%2Aw%5E2+=+root%283%2C4%29%2A%281-w%29 ALIGN=MIDDLE  ALT=\"root%283%2C+7%29%2Aw%5E2+=+root%283%2C4%29%2A%281-w%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, after taking cube root of both sides.\r<BR>\n" );
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document.write( "===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C+7%29%2Aw%5E2+%2B+root%283%2C4%29%2Aw+-+root%283%2C4%29+=+0%29 ALIGN=MIDDLE  ALT=\"root%283%2C+7%29%2Aw%5E2+%2B+root%283%2C4%29%2Aw+-+root%283%2C4%29+=+0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+%28-root%283%2C4%29+%2B-+sqrt%28+root%283%2C16%29%2B4root%283%2C28%29%29%29%2F%282%2Aroot%283%2C7%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+%28-root%283%2C4%29+%2B+sqrt%28+root%283%2C16%29%2B4root%283%2C28%29%29%29%2F%282%2Aroot%283%2C7%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >, since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+0+%3C=+w+%3C=+1 ALIGN=MIDDLE  ALT=\"+0+%3C=+w+%3C=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "&#8776; 0.5860760201.\r<BR>\n" );
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document.write( "Now to the left of 0.5860760201, z' < 0, so z is decreasing on [0, 0.5860760201).\r<BR>\n" );
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document.write( "To the right of 0.5860760201, z' > 0, so z is increasing on (0.5860760201,1].\r<BR>\n" );
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document.write( "Hence there is a local minimum at w = 0.5860760201.\r<BR>\n" );
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document.write( "We only need to check the other critical points where the derivative doesn't exist, namely at w = 0 and w = 1, <BR>\n" );
document.write( "to determine the other possible extrema.\r<BR>\n" );
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document.write( "At w = 0, z = 1.<BR>\n" );
document.write( "At w = 1, z = 1.\r<BR>\n" );
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document.write( "Therefore the maximum of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%5E8%28x%29%2Bcos%5E14%28x%29 ALIGN=MIDDLE  ALT=\"sin%5E8%28x%29%2Bcos%5E14%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%281%29 ALIGN=MIDDLE  ALT=\"highlight%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, while the minimum is \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.5860760201%5E7+%2B+%281-0.5860760201%29%5E4+=+highlight%280.05310578625%29 ALIGN=MIDDLE  ALT=\"0.5860760201%5E7+%2B+%281-0.5860760201%29%5E4+=+highlight%280.05310578625%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, approximately.<BR>\n" );
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