SOLUTION: 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
F(x) = 2x^2-1/2x-3/2
F(x) = 3
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Question 1051745: 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
F(x) = 2x^2-1/2x-3/2
F(x) = 3x^2-2x+4
F(x) =x^2+5x
2) Find two positive numbers such that their sum is 20 and their product is a maximum.
3) A rectangle has a perimeter of 100 meters. What are the dimensions of the sides if the area is a maximum?
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?
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?
Answer by Boreal(15235) (Show Source): You can put this solution on YOUR website!
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.
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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.
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x^2+5x: maximum value is infinite. Vertex is at x=-5/2 where f(x)=6.25-12.5=-6.25
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x and (20-x)
product is 20x-x^2
-x^2+20 has a maximum where x=10 (-b/2a) and f(x)=100
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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.
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.
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piece for circle is x
The perimeter x=2*pi*r, so r=x/2*pi and area is pi*x^2/4*pi^2=x^2/4pi
The square is has perimeter 20-x, so each side is (5-0.25x) and area is 25-0.5x+0.0625x^2.
The sum of both is (1/4pi)*(x^2)+0.0625x^2-0.5x+25
That is (0.0625+(1/4pi))x^2-0.5x+25
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.
f(x)=(0.0625+(1/4pi)(3.10)-0.1474+25=0.1420(3.10)-0.1474+25=25.29 in^2
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Revenue is (300-10x)(6+0.25x)=1800+15x-2.5x^2
The vertex has an x-value of -b/2z= -15/-5=3
The revenue is maximized at 270 people paying $6.75=$1822.50. Increasing the price to $6.75 is the maximum.
Check
280*6.50=$1820
260*$7=$1820
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