Chapter 18: Optimization Techniques
MULTIPLE CHOICE
1. 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.
2. 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.
3. 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.
4. 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.
5. 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.
6. 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.
7. 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.
8. The slope of a straight line is:
a.
positive.
b.
negative.
c.
zero.
d.
constant.
e.
nonconstant.
9. The constant rule of differentiation is:
a.
Y = a + bX dY/dX = a + b.
b.
Y = a + bX dY/dX = a.
c.
Y = a + bX dY/dX = b.
d.
Y = a + bX dY/dX = ab.
e.
Y = a + bX dY/dX = ab.
10. 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.
11. If Y = –2 + X + 32X3, then dY/dX is:
a.
1 + 96X3.
b.
–1 + 96X2.
c.
1 + 96X2.
d.
96X2.
e.
X + 32X3.
12. If Y = a + bX + cXd, then dY/dX is:
a.
a + bX + cXd.
b.
b – 1 + (c – 1)Xd – 1.
c.
b + (d – 1)(c – 1)Xd – 1.
d.
b + cdXd – 1.
e.
bX + cXd.
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.
14. 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.
15. If Y = 3X(X3), then d2Y/dX 2 is:
a.
6X 2.
b.
12X3.
c.
18X 2.
d.
24X 2.
e.
36X 2.
16. If Y = aXb(c + Xd), then dY/dX is:
a.
abXb – 1(c + X)d + aXb dXd – 1.
b.
abXb – 1(c + X)d + aXb(d – 1)Xd.
c.
a(b – 1)Xb(c + Xd) + aXb(d – 1)Xd.
d.
abXb – 1 dXd – 1.
e.
a(b – 1)Xb(c + Xd).
17. If Y = 21X1/3(25 + X4), then dY/dX is:
a.
7X –2/3(25 + X4) + 84X10/3.
b.
7X2/3(25 + X4) + 84X10/3.
c.
7X –1/3(25 + X4) + 84X10/3.
d.
7X1/3(25 + X4) + 84X10/3.
e.
7X1/3(25 + X4).
18. If Y = aX / (b + Xc), then dY/dX is:
a.
[a(b + Xc) – acXc] / (b + Xc)2.
b.
a(b + Xc) + acXc – 1.
c.
[a(b + Xc) – acXc – 1] / (b + Xc).
d.
[a(b + Xc) + acXc – 1] / (b + Xc)2.
e.
a(b + Xc) / (b + Xc)2.
19. 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.
20. 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).
21. 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).
22. 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.
23. The power rule of differentiation is:
a.
Y = aXb dY/dX = (b – 1)aXb – 1.
b.
Y = aXb dY/dX = (a – 1)bXa – 1.
c.
Y = aXb dY/dX = (a – 1)aXa – 1.
d.
Y = aXb dY/dX = (b – 1)bXb – 1.
e.
Y = aXb dY/dX = baXb – 1.
24. The quotient rule of differentiation is:
a.
Y = U(X) / W(X) dY/dX = (W dU/dX – U dW/dX) / W 2.
b.
Y = U(X) / W(X) dY/dX = (W dU/dX – U dW/dX) / W.
c.
Y = U(X) / W(X) dY/dX = [(W dU/dX) / (U dW/dX)] / W 2.
d.
Y = U(X) / W(X) dY/dX = [(W dU/dX)(U dW/dX)] / W 2.
e.
Y = U(X) / W(X) dY/dX = [(W dU/dX)(U dW/dX)] / W.
25. The product rule of differentiation is:
a.
Y = U(X)W(X) dY/dX = (U dW/dX)(W dU/dX).
b.
Y = U(X)W(X) dY/dX = (W dW/dX) / (U dU/dX).
c.
Y = U(X)W(X) dY/dX = (U dW/dX) – (W dU/dX).
d.
Y = U(X)W(X) dY/dX = (U dW/dX) + (W dU/dX).
e.
Y = U(X)W(X) dY/dX = (W dW/dX) + (U dU/dX).
26. 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.
27. The chain rule of differentiation is:
a.
Y = U(W(X)) dY/dX = dY/dX dW/dX.
b.
Y = U(W(X)) dY/dX = dU/dW dW/dX.
c.
Y = U(W(X)) dY/dX = dU/dX dW/dX.
d.
Y = U(W(X)) dY/dX = dW/dU dU/dX.
e.
Y = U(W(X)) dY/dX = dU/dU dU/dX.
28. 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.
29. 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.
30. 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.
31. Too Much Fun (TMF) sells board games for the discerning student. It estimates that its total cost of
sales is TC = Q + 27Q1/3. At 27 units of output, TMF’s marginal cost is:
a.
$2.
b.
$29.
c.
$27.
d.
$30.
e.
$28.
32. 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.
33. Al’s Authentic Anklettes sells ankle bracelets. Their demand and costs are given by QD = 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.
34. 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.
35. 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.
36. 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.
37. Slim’s Shoe Repair has determined that its total cost of resoling shoes is TC = Q2 + 16Q1/2. At 16 units
of output, Slim’s marginal cost is:
a.
$324.
b.
$64.
c.
$256.
d.
$34.
e.
$2.
38. 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.
39. Bolan’s Fabric Shop sells discount material. Its demand and cost functions are QD = 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.
40. 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.
41. 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.
42. 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.
43. 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.
44. 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.
45. 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.
46. 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.
47. When using the Lagrangian technique for solving a constrained cost-minimization problem, the
Lagrangian multiplier 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.
48. 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.
49. 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.
50. 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.
51. 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.
52. 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.
53. 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.
54. 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.
55. 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.