document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1051745>Question 1051745</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1) Determine if f has a maximum value or a minimum value, te value of x at which the maximum or minimum occurs and the maximum or minimum value of f<BR>\n" );
document.write( "F(x) = 2x^2-1/2x-3/2<BR>\n" );
document.write( "F(x) = 3x^2-2x+4<BR>\n" );
document.write( "F(x) =x^2+5x\r<BR>\n" );
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document.write( "2) Find two positive numbers such that their sum is 20 and their product is a maximum. \r<BR>\n" );
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document.write( "3) A rectangle has a perimeter of 100 meters. What are the dimensions  of the sides if the area is a maximum? \r<BR>\n" );
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document.write( "4)A piece of wire 20 inches long is to be cut into two pieces, one of which will be bent into a circle and the other into a square.  How long should each piece be to minimize the sum of the areas? \r<BR>\n" );
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document.write( "5) A movie theater finds that 300 people attend each performance at the current  rate of $6 and that attendance decreases by 10 persons for each 25 cent increase in price.  What increase yields the greatest gross revenue?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #667280 by </B><A HREF=/tutors/aboutme.mpl?userid=Boreal><B>Boreal(15235)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Boreal><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=Boreal><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=667280\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> f(x)=2x^2-(1/2)x-3/2.  The maximum value is infinite. The minimum value is at the vertex, where the x value is -b/2a=1/2 divided by 2 or x=1/4.  F(1/4)=(1/8)-1/8-3/2 or -3/2.  Graph.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-5%2C5%2C-5%2C5%2C2x%5E2-%281%2F2%29x-3%2F2%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-5%2C5%2C-5%2C5%2C2x%5E2-%281%2F2%29x-3%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "f(x)=3x^2-2x+4. Maximum value is infinite, and vertex is at x=2/6 or x=1/3. f(1/3) is (1/3)-(2/3)+4=11/3. That is the minimum value.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-5%2C5%2C-5%2C5%2C3x%5E2-2x%2B4%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-5%2C5%2C-5%2C5%2C3x%5E2-2x%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "x^2+5x: maximum value is infinite.  Vertex is at x=-5/2 where f(x)=6.25-12.5=-6.25<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-5%2C5%2C-10%2C5%2Cx%5E2%2B5x%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-5%2C5%2C-10%2C5%2Cx%5E2%2B5x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "x and (20-x)<BR>\n" );
document.write( "product is 20x-x^2<BR>\n" );
document.write( "-x^2+20 has a maximum where x=10 (-b/2a) and f(x)=100<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-10%2C12%2C-10%2C125%2C-x%5E2%2B20x%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-10%2C12%2C-10%2C125%2C-x%5E2%2B20x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "These are squares.  Call the semi-perimeter 50, so that one side is x and the other is 50-x. That will give a perimeter of 100.<BR>\n" );
document.write( "The area is x(50-x)=-x^2+50x.  The maximum is where -b/2a is x which is -50/-2=25. A rectangle with perimeter 100 and side length of 25 is a square.<BR>\n" );
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document.write( "piece for circle is x<BR>\n" );
document.write( "The perimeter x=2*pi*r, so r=x/2*pi and area is pi*x^2/4*pi^2=x^2/4pi<BR>\n" );
document.write( "The square is has perimeter 20-x, so each side is (5-0.25x) and area is 25-0.5x+0.0625x^2.<BR>\n" );
document.write( "The sum of both is (1/4pi)*(x^2)+0.0625x^2-0.5x+25<BR>\n" );
document.write( "That is (0.0625+(1/4pi))x^2-0.5x+25<BR>\n" );
document.write( "The minimum x-value is -b/2a, which is 0.5/0.125+(1/2pi). Numerically, this is 1.76 in.  That is the piece for the circle.  The square will be formed by a piece 18.24 in.  <BR>\n" );
document.write( "f(x)=(0.0625+(1/4pi)(3.10)-0.1474+25=0.1420(3.10)-0.1474+25=25.29 in^2<BR>\n" );
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document.write( "Revenue is (300-10x)(6+0.25x)=1800+15x-2.5x^2<BR>\n" );
document.write( "The vertex has an x-value of -b/2z= -15/-5=3<BR>\n" );
document.write( "The revenue is maximized at 270 people paying $6.75=$1822.50.  Increasing the price to $6.75 is the maximum.<BR>\n" );
document.write( "Check<BR>\n" );
document.write( "280*6.50=$1820<BR>\n" );
document.write( "260*$7=$1820<BR>\n" );
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