Find the critical point of the function f(x, y) = 6 + 7x – 2x2 + 8y + 7y2
8
14
This critical point is a: Saddle
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Suppose that f(x, y) = 4x4 + 4y4 – 2xy
Then the minimum value is
-
8
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Find and classify the critical points of z =
(22 – 6x) (y2 – 5y)
Local maximums at:
(3,3)
Local minimums at: DNE
Saddle points at: (0,0),(0,5),(6,0),(6,5)
For each of the fields above, enter an ordered pair (x, y) (or a comma-separated list of ordered
pairs) where the max/min/saddle occurs. If there are none enter DNE.
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A company manufactures 2 models of MP3 players. Let x represent the number (in millions) of the first
model made, and let y represent the number (in millions) of the second model made.
The company's revenue can be modeled by the equation
R(x, y)
90x + 1807 – 2x2 – 4y2 – xy
Find the marginal revenue equations
Rz(x, y)
=
90 – 4x – y
Ry(x,y)
=
180 – 8y - 2
We can achieve maximum revenue when both partial derivatives are equal to zero. Set Rx 0 and
Ry O and solve as a system of equations to the find the production levels that will maximize revenue.
Revenue will be maximized when:
X =
17.4194
y =
20.3226
O-
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An open-top rectangular box is being constructed to hold a volume of 350 in”. The base of the box is made
from a material costing 6 cents/in2. The front of the box must be decorated, and will cost 9 cents/in2. The
remainder of the sides will cost 4 cents/in?.
Find the dimensions that will minimize the cost of constructing this box.
Front width: 6.59758
in.
Depth: 10.72106
om in.
Height: | 4.94818
in.
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Find the maximum and minimum values of the function f(x, y)
= exy
subject to x3 + y2 = 54
Please show your answers to at least 4 decimal places. Enter DNE if the value does not exist.
Maximum value: 8103.0839
0
Minimum value: DNE
Ꮕ
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Suppose a Cobb-Douglas Production function is given by the following:
P(L,K) = 700.75 0.25
where L is units of labor, K is units of capital, and P(L, K) is total units that can be produced with this
labor/capital combination. Suppose each unit of labor costs $400 and each unit of capital costs $800.
Further suppose a total of $192,000 is available to be invested in labor and capital (combined).
A) How many units of labor and capital should be "purchased" to maximize production subject to your
budgetary constraint?
Units of labor, L
Units of capital, K
B) What is the maximum number of units of production under the given budgetary conditions? (Round your
answer to the nearest whole unit.)
Max production
units
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Suppose a Cobb-Douglas Production function is given by the function: P(L, K).
=
2440.8 K0.2
Furthemore, the cost function for a facility is given by the function:C(L, K) = 300L + 500K
Suppose the monthly production goal of this facility is to produce 12,000 items. In this problem, we will
assume L represents units of labor invested and K represents units of capital invested, and that you can
invest in tenths of units for each of these. What allocation of labor and capital will minimize total
production Costs?
Units of Labor L =
(Show your answer is exactly 1 decimal place)
Units of Capital K
=
(Show your answer is exactly 1 decimal place)
Also, what is the minimal cost to produce 12,000 units? (Use your rounded values for L and K from above
to answer this question.)
The minimal cost to produce 12,000 units is $
Hint:
1. Your constraint equation involves the Cobb Douglas Production function, not the Cost function.
2. When finding a relationship between L and K in your system of equations, remember that you will
want to eliminate to get a relationship between L and K.
3. Round your values for L and K to one decimal place (tenths).
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Find the critical point of the function f(2, y) = 2 + 6x + 7x2 - 5y - 4y?
This critical point is a v Select an answer
Saddle
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Suppose that f(x, y) = x4 + y4 - 2xy
Then the minimum value is
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An open-top rectangular box is being constructed to hold a volume of 350 in. The base of the box is made
from a material costing 7 cents/in?. The front of the box must be decorated, and will cost 10 cents/in?
The remainder of the sides will cost 2 cents/in?.
Find the dimensions that will minimize the cost of constructing this box.
Front width:
in.
Depth:
in.
Height:
in.
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Find the maximum and minimum values of the function f($, y) = efy subject to 2 + y = 54.
Hint: one of these values exists, and the other one doesn't exist.
Maximum value:
Please show your answers to at least 2 decimal places. Enter DNE if the value does not exist.
Minimum value:
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Suppose a Cobb-Douglas Production function is given by the function: P(L,K) = 24LSKI
Furthemore, the cost function for a facility is given by the function:C(L,K) = 100L + 500K
Suppose the monthly production goal of this facility is to produce 5,000 items. In this problem, we will
assume L represents units of labor invested and K represents units of capital invested, and that you can
invest in tenths of units for each of these. What allocation of labor and capital will minimize total
production Costs?
Units of Labor L.
(Show your answer is exactly 1 decimal place)
Units of Capital K
(Show your answer is exactly 1 decimal place)
Also, what is the minimal cost to produce 5,000 units? (Use your rounded values for L and K from above to
answer this question.)
The minimal cost to produce 5,000 units is $
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