Deck 18: Optimization Techniques

Question 1

If Y = 3 / X,then d²Y/dX ² is:

A) -6 / X ³.
B) -3 / X ².
C) 6 / X ².
D) 6 / X ³.
E) 6X ³.

Answer 1

D

Question 2

The first derivative of total profit with respect to quantity is:

A) average revenue.
B) marginal revenue.
C) marginal profit.
D) average profit.
E) total profit.

Answer 2

C

Question 3

The slope of a straight line is:

A) positive.
B) negative.
C) zero.
D) constant.
E) nonconstant.

Answer 3

D

Question 4

When average profit is increasing with increases in output,marginal profit must be:

A) increasing.
B) less than average profit.
C) greater than average profit.
D) decreasing.
E) constant.

Question 5

If Y = -2 + X + 32X³,then dY/dX is:

A) 1 + 96X³.
B) -1 + 96X².
C) 1 + 96X².
D) 96X².
E) X + 32X³.

Question 6

The constant rule of differentiation is:

A) Y = a + bX $\rarr$ dY/dX = a + b
B) Y = a + bX $\rarr$ dY/dX = a
C) Y = a + bX $\rarr$ dY/dX = b
D) Y = a + bX $\rarr$ dY/dX = ab
E) Y = a + bX $\rarr$ dY/dX = ab

Question 7

Whenever average profit is declining with increases in output,marginal

A) revenue is less than marginal cost.
B) revenue is greater than marginal cost.
C) revenue is equal to marginal cost.
D) profit is less than average profit.
E) profit is negative.

Question 8

If Y = aX^b(c + X^d),then dY/dX is:

A) abX^{b - 1}(c + X)^d + aX^b dX^{d - 1}.
B) abX^{b - 1}(c + X)^d + aX^b(d - 1)X^d.
C) a(b - 1)X^b(c + X^d) + aX^b(d - 1)X^d.
D) abX^{b - 1} dX^{d - 1}.
E) a(b - 1)X^b(c + X^d).

Question 9

If Y = aX / (b + X^c),then dY/dX is:

A) [a(b + X^c) - acX^c] / (b + X^c)².
B) a(b + X^c) + acX^{c - 1}.
C) [a(b + X^c) - acX^{c - 1}] / (b + X^c).
D) [a(b + X^c) + acX^{c - 1}] / (b + X^c)².
E) a(b + X^c) / (b + X^c)².

Question 10

Marginal profit is maximized when:

A) average profit is equal to marginal profit.
B) total profit is maximized.
C) average profit is increasing.
D) average profit is maximized.
E) average profit is decreasing.

Question 11

Whenever average profit is less than marginal profit:

A) average profit declines with increases in output.
B) marginal profit decreases with increases in output.
C) marginal profit increases with increases in output.
D) average profit is maximized.
E) average profit increases with increases in output.

Question 12

If Y = 12 - 6X + 8X ^-1/2,then dY/dX is:

A) -6 - 4X ^-3/2.
B) -4X ^-3/2.
C) 6 - 4X ^-3/2.
D) -6 - 4X ^-1/2.
E) -6 - X ^-3/2.

Question 13

If Y = 12 - 6X + 3X³,then d²Y/dX² is:

A) -6 + 9X ².
B) 9X.
C) -6 + 18X ².
D) 18X.
E) none of the above.

Question 14

If Y = (X - 3)² / (9 + 12X - X⁴),then dY/dX is:

A) [2(9 + 12X - X⁴)(X - 3) + (12 - 4X³)(X - 3)²] / (9 + 12X - X⁴)².
B) [2(9 + 12X - X⁴)(X - 3) - (12 - 4X³)(X - 3)²] / (9 + 12X - X⁴)².
C) [2(9 + 12X - X⁴)(X - 3) + (12 - 4X³)(X - 3)²](9 + 12X - X⁴)².
D) [2(9 + 12X - X⁴)(X - 3) - (12 - 4X³)(X - 3)²](9 + 12X - X⁴)².
E) [2(9 + 12X - X⁴)(X - 3) - (12 - 4X³)(X - 3)²] / (9 + 12X - X⁴).

Question 15

If Y = (10 - X ²)^1/2,then dY/dX is:

A) -X(10 - X ²) ^-1/2.
B) 1/2 (10 - X ²) ^-1/2.
C) -X(10 - X ²)^1/2.
D) 1/2(10 - X ²) ^-1/2.
E) none of the above.

Question 16

Total profit is maximized when:

A) marginal profit equals average profit.
B) marginal profit equals zero.
C) average profit equals zero.
D) average profit is maximized.
E) marginal profit is greater than average profit.

Question 17

Average profit is maximized when:

A) average profit is equal to marginal profit.
B) total profit is maximized.
C) marginal profit is increasing.
D) marginal profit is maximized.
E) marginal profit is minimized.

Question 18

If Y = a + bX + cX^d,then dY/dX is:

A) a + bX + cX^d.
B) b - 1 + (c - 1)X^{d - 1}.
C) b + (d - 1)(c - 1)X^{d - 1}.
D) b + cdX^{d - 1}.
E) bX + cX^d.

Question 19

If Y = 3X(X³),then d²Y/dX ² is:

A) 6X ².
B) 12X³.
C) 18X ².
D) 24X ².
E) 36X ².

Question 20

If Y = 21X^1/3(25 + X⁴),then dY/dX is:

A) 7X ^-2/3(25 + X⁴) + 84X^10/3.
B) 7X^2/3(25 + X⁴) + 84X^10/3.
C) 7X ^-1/3(25 + X⁴) + 84X^10/3.
D) 7X^1/3(25 + X⁴) + 84X^10/3.
E) 7X^1/3(25 + X⁴).

Question 21

Too Much Fun (TMF)sells board games for the discerning student.It estimates that its total cost of sales is TC = Q + 27Q^1/3.At 27 units of output,TMF's marginal cost is:

A) $2.
B) $29.
C) $27.
D) $30.
E) $28.

Question 22

Bolan's Fabric Shop sells discount material.Its demand and cost functions are Q_D = 40 - 2P and TC = 0.5Q².Its profit-maximizing price is:

A) P = $5.
B) P = $10.
C) P = $15.
D) P = $20.
E) P = $0.

Question 23

Fox's Fine Furs (FFF)estimates that its total cost of production is TC = 125 + 100Q + 25Q².Furs sell for $1,100 each.To maximize profits,FFF should sell:

A) 8 furs.
B) no furs.
C) 20 furs.
D) 38 furs.
E) 22 furs.

Question 24

If marginal revenue exceeds marginal costs,to increase profits a firm should:

A) reduce output.
B) increase output.
C) hold output constant.
D) increase marginal revenue.
E) reduce marginal cost.

Question 25

If Y = X³(5 + X ²)⁴,then dY/dX is:

A) 3X ²(5 + X ²)⁴ + 8X³(5 + X ²)³.
B) 3X ²(5 + X ²)⁴ + 8X⁴(X ²)³.
C) 3X ²(5 + X ²)³.
D) 3X ²(5 + X ²)⁴ + 8X⁴(5 + X ²)³.
E) 3X ²(5 + X ²)⁴ + 8X⁴(5 + X ²)².

Question 26

The chain rule of differentiation is:

A) Y = U(W(X)) $\rarr$ dY/dX = dY/dX dW/dX.
B) Y = U(W(X)) $\rarr$ dY/dX = dU/dW dW/dX.
C) Y = U(W(X)) $\rarr$ dY/dX = dU/dX dW/dX.
D) Y = U(W(X)) $\rarr$ dY/dX = dW/dU dU/dX.
E) Y = U(W(X)) $\rarr$ dY/dX = dU/dU dU/dX.

Question 27

The product rule of differentiation is:

A) Y = U(X)W(X) $\rarr$ dY/dX = (U dW/dX)(W dU/dX).
B) Y = U(X)W(X) $\rarr$ dY/dX = (W dW/dX) / (U dU/dX).
C) Y = U(X)W(X) $\rarr$ dY/dX = (U dW/dX) - (W dU/dX).
D) Y = U(X)W(X) $\rarr$ dY/dX = (U dW/dX) + (W dU/dX).
E) Y = U(X)W(X) $\rarr$ dY/dX = (W dW/dX) + (U dU/dX).

Question 28

Al's Authentic Anklettes sells ankle bracelets.Their demand and costs are given by Q^D = 100 - P and TC = 100 + 10Q.The profit-maximizing level of output is:

A) 45 ankle bracelets.
B) 47.5 ankle bracelets.
C) 90 ankle bracelets.
D) 10 ankle bracelets.
E) no ankle bracelets.

Question 29

The power rule of differentiation is:

A) Y = aX^b $\rarr$ dY/dX = (b - 1)aX^{b - 1}.
B) Y = aX^b $\rarr$ dY/dX = (a - 1)bX^{a - 1}.
C) Y = aX^b $\rarr$ dY/dX = (a - 1)aX^{a - 1}.
D) Y = aX^b $\rarr$ dY/dX = (b - 1)bX^{b - 1}.
E) Y = aX^b $\rarr$ dY/dX = baX^{b - 1}.

Question 30

A function of one argument is maximized when the first derivative:

A) is zero and the second derivative is positive.
B) is positive and the second derivative is negative.
C) is zero and the second derivative is negative.
D) is negative and the second derivative is positive.
E) and the second derivative are both zero.

Question 31

Kenny's Cartage hauls crushed stone for $15 a ton and has total costs given by TC = 100 + 5X + X².The profit-maximizing level of output is:

A) 5 tons.
B) 2.1 tons.
C) 10 tons.
D) 20 tons.
E) 0 tons.

Question 32

NotAlligator Briefcases estimates that its total cost of producing vinyl bags is TC = 65 + 9 / Q + 5Q².At 3 units of output,NotAlligator's marginal cost is:

A) $113.
B) $29.
C) $27.
D) $119.
E) $9.

Question 33

If Y = U(X)W(X),then dY/dX is:

A) U dW/dX - W dU/dX.
B) U dW/dX + W dU/dX.
C) (W dW/dX) / (U dU/dX).
D) (U dW/dX)(W dU/dX).
E) W dW/dX + U dU/dX.

Question 34

If Y = 3X / (3X + X ²),then dY/dX is:

A) [(3 + 2X)3X + 3(3X + X ²)] / (3X + X ²)².
B) (3 + 2X)3X / (3X + X²)².
C) -3 / (3 + X)².
D) [(3 + 2X) - 3(3X + X ²)] / (3X + X ²)².
E) (3 + 2X)3X / (3X + X²).

Question 35

The quotient rule of differentiation is:

A) Y = U(X) / W(X) $\rarr$ dY/dX = (W dU/dX - U dW/dX) / W ².
B) Y = U(X) / W(X) $\rarr$ dY/dX = (W dU/dX - U dW/dX) / W.
C) Y = U(X) / W(X) $\rarr$ dY/dX = [(W dU/dX) / (U dW/dX)] / W ².
D) Y = U(X) / W(X) $\rarr$ dY/dX = [(W dU/dX)(U dW/dX)] / W ².
E) Y = U(X) / W(X) $\rarr$ dY/dX = [(W dU/dX)(U dW/dX)] / W.

Question 36

Murdock Glass sells stained glass panes.Its profit is given by p = -500 + 100X - X².The profit-maximizing level of output is:

A) X = 50.
B) X = 100.
C) X = 200.
D) X = 300.
E) X = 600.

Question 37

A function of one argument is minimized when the first derivative is:

A) zero and the second derivative is positive.
B) positive and the second derivative is negative.
C) zero and the second derivative is negative.
D) negative and the second derivative is positive.
E) zero and the second derivative is zero.

Question 38

Shag Express,a retailer of lamps,has determined that its total cost of retailing lamps is TC = 200 + 10Q + 5Q².At 10 units of output,the firm's marginal cost is:

A) $110.
B) $100.
C) $800.
D) $230.
E) $10.

Question 39

Slim's Shoe Repair has determined that its total cost of resoling shoes is TC = Q² + 16Q^1/2.At 16 units of output,Slim's marginal cost is:

A) $324.
B) $64.
C) $256.
D) $34.
E) $2.

Question 40

Gibbon's Restaurant finds that it sells more pizzas when it advertises according to S = 20 + 4A - 0.5A²,where S is sales and A is advertising expenditure.The sales-maximizing level of advertising is:

A) A = $2.
B) A = $4.
C) A = $6.
D) A = $8.
E) A = $12.

Question 41

Campbell's sells used trailers,U,and new trailers,N.Its profits are given by p = 100N + 68U - 5N² - 5U² - 2NU.The profit-maximizing combination of trailers for Campbell's is:

A) N = 9 and U = 5.
B) N = 7 and U = 7.
C) N = 5 and U = 9.
D) N = 13 and U = 0.
E) N = 9 and U = 9.

Question 42

Using the Lagrangian multiplier technique,you allocate your $1,000,000 advertising budget between four media markets so as to maximize your profits.Your assistant informs you that the Lagrangian multiplier is equal to -$0.50.From this you conclude that:

A) you are allocating your budget across markets optimally.
B) advertising is worthless.
C) another dollar of advertising would increase profits $0.50.
D) another dollar of advertising would decrease profits $0.50.
E) you could earn more profit with a larger advertising budget.

Question 43

If marginal revenue is less than marginal cost at every level of output,a profit-maximizing firm should:

A) produce when the difference between marginal revenue and marginal cost is greatest.
B) produce when total revenue is maximized.
C) produce when the difference between total revenue and marginal cost is maximized.
D) produce when the difference between average revenue and average cost is equal to 1.
E) not produce any output.

Question 44

Carmen's Detective Agency can use apprentice detectives,A,or experienced detectives,E.The cost of completing a job is C = 100 + A² + E² - 2AE.If a job requires a total of six detectives of either or both types,the cost-minimizing combination of detectives is:

A) A = 3 and E = 3.
B) A = 2 and E = 4.
C) A = 4 and E = 2.
D) A = 6 and E = 0.
E) A = 0 and E = 6.

Question 45

Sally sells sandals.She can advertise on radio,A₁,or on television,A₂.Profits depend on advertising according to p = 100 + 10A₁ + 20A₂ - A²₁ - A²₂ + 0.5A₁A₂.The profit-maximizing levels of radio and television advertising are:

A) A₁ = $8 and A₂ = $12.
B) A₁ = $12 and A₂ = $12.
C) A₁ = $8 and A₂ = $8.
D) A₁ = $12 and A₂ = $8.
E) A₁ = $10 and A₂ = $10.

Question 46

The second derivative of the total profit function is:

A) average profit.
B) marginal profit.
C) the slope of the average profit function.
D) the slope of the marginal profit function.
E) the slope of the total profit function.

Question 47

You must produce 200 records this week.Using the Lagrangian technique,you determine the profit-maximizing combination of pink vinyl and blue vinyl records and find that the Lagrangian multiplier is $2.50.From this you conclude that:

A) the marginal revenue is about $2.50.
B) the average revenue is about $2.50.
C) the average revenue exceeds average cost by about $2.50.
D) the marginal revenue exceeds marginal cost by about $2.50.
E) consumers prefer pink vinyl to blue vinyl.

Question 48

You only have 12 ovens in which to bake over 200 specialty pastries.Subject to the oven constraint,you determine the profit-maximizing quantities of each pastry to produce and find that the Lagrangian multiplier is equal to 0.From this you conclude that you:

A) should be producing fewer types of pastry.
B) should be producing more types of pastry.
C) should purchase additional ovens.
D) are effectively unconstrained with 12 ovens.
E) should purchase 2 more ovens.

Question 49

Wilma's Car Repair can repair cars using kryptonite bolts,K,or lithium bolts,L,as long as it uses 10 bolts in toto.The cost of repairing a car is TC = K² + L² - KL.The cost-minimizing combination of kryptonite and lithium bolts is:

A) K = 6 and L = 4.
B) K = 4 and L = 6.
C) K = 7 and L = 3.
D) K = 3 and L = 7.
E) K = 5 and L = 5.

Question 50

Maximum profit occurs wherever:

A) the slope of the total revenue function equals marginal revenue.
B) the slope of the total revenue function equals marginal cost.
C) the slope of the total revenue function is maximized.
D) the total revenue is maximized.
E) none of the above.

Question 51

When using the Lagrangian technique for solving a constrained cost-minimization problem,the Lagrangian multiplier $\lambda$ is:

A) the optimal level of cost.
B) the minimized marginal cost.
C) the minimized total cost.
D) the maximized profit.
E) equal to zero.

Question 52

Campbell's sells used trailers,U,and new trailers,N.Its profits are given by p = 100N + 68U - 5N² - 5U² - 2NU.Campbell's maximum profit is:

A) $455.
B) $588.
C) $620.
D) $495.
E) $640.

Question 53

Sally can advertise on radio,A₁,or on television,A₂,as long as she spends no more than $10.Profits depend on her advertising according to p = 100 + 10A₁ + 20A₂ - A²₁ - A²₂ + 0.5A₁A₂.The constrained profit-maximizing levels of radio and television advertising are:

A) A₁ = $3 and A₂ = $7.
B) A₁ = $7 and A₂ = $3.
C) A₁ = $10 and A₂ = $0.
D) A₁ = $0 and A₂ = $10.
E) A₁ = $5 and A₂ = $5.

Question 54

Incremental revenues and incremental costs are used when:

A) derivatives are too complicated.
B) revenues and costs are unknown.
C) managerial decisions involve substantial changes.
D) a decision is to be presented to a nontechnical audience.
E) managerial decisions involve infinitesimal changes.

Question 55

Instructed to choose the combination of inputs that minimizes the cost of producing 2,000 pan-head screws,your assistant returns with the startling news that the Lagrangian multiplier is $0.10.From this you conclude that:

A) costs have not been minimized.
B) the average cost of producing screws at 2,000 units is $0.10.
C) the marginal cost of producing screws at 2,000 units is $0.10.
D) marginal cost equals average cost.
E) the price of the screws must be $0.10.