document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1027573>Question 1027573</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let f be a function such that f(0) = 2 and f'(x)<=6 for -10<=x<=10.\r<BR>\n" );
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document.write( "A) What is the maximum possible value of f(4)?<BR>\n" );
document.write( "B) What is the maximum possible value of f(2)?<BR>\n" );
document.write( "C) Can f(5) be negative? Can f(5) = 0? Why or Why not?<BR>\n" );
document.write( "D) Can f(5) = 31? Can f(5) = 35? Why or why not?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #642786 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=642786\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> By the mean value theorem,\r<BR>\n" );
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document.write( "there is a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=+alpha+%3C=+x ALIGN=MIDDLE  ALT=\"0%3C=+alpha+%3C=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" > such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+f%280%29+%2B+%28df%28alpha%29%2Fdx%29x ALIGN=MIDDLE  ALT=\"f%28x%29+=+f%280%29+%2B+%28df%28alpha%29%2Fdx%29x\"   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=f%28x%29+%3C=+f%280%29+%2B+6x+=+2+=+6x ALIGN=MIDDLE  ALT=\"f%28x%29+%3C=+f%280%29+%2B+6x+=+2+=+6x\"   DISABLED_style= \"max-width:90%; height:auto;\" >  because  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=df%28x%29%2Fdx+%3C=+6 ALIGN=MIDDLE  ALT=\"df%28x%29%2Fdx+%3C=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" > for all x in [-10,10].\r<BR>\n" );
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document.write( "A)  ==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%284%29+%3C=+2+%2B+6%2A4+=+26 ALIGN=MIDDLE  ALT=\"f%284%29+%3C=+2+%2B+6%2A4+=+26\"   DISABLED_style= \"max-width:90%; height:auto;\" > ==> maximum possible value of f(4) is 26.\r<BR>\n" );
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document.write( "B)  ==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%282%29+%3C=+2+%2B+6%2A2+=+14 ALIGN=MIDDLE  ALT=\"f%282%29+%3C=+2+%2B+6%2A2+=+14\"   DISABLED_style= \"max-width:90%; height:auto;\" > ==> maximum possible value of f(2) is 14.\r<BR>\n" );
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document.write( "C)  Can f(5) be negative? YES. The derivative in the neighborhood around x = 5 can be negative enough (and hence the graph fall drastically enough) so as to warrant the graph of of f(x) to cross the x-axis before x = 5.  (Remember, f'(x) is less than or equal to 6 in [-10,10].)\r<BR>\n" );
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document.write( "Can f(5) = 0? YES, by a reasoning very similar to the preceding paragraph.\r<BR>\n" );
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document.write( "D) Can f(5) = 31? YES.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%285%29+%3C=+2+%2B+6%2A5+=+32 ALIGN=MIDDLE  ALT=\"f%285%29+%3C=+2+%2B+6%2A5+=+32\"   DISABLED_style= \"max-width:90%; height:auto;\" > ==> maximum possible value of f(5) is 32.\r<BR>\n" );
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document.write( "Can f(5) = 35?  NO. The highest f(5) can get is 32 </SPAN>\n" );
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