506
CHAPTER 7
Multivariable Calculus
Step 2 Compute A = Mxx 16, 42, B = Mxy 16, 42, and C = Myy 16, 42:
Mxx 1x, y2 =
2 inches
Mxy 1x, y2 = 1;
4 inches
6 inches
288
;
x3
Myy 1x, y2 =
Figure 6
192
;
y3
so
so
so
A = Mxx 16, 42 =
288
216
C = Myy 16, 42 =
192
64
B = Mxy 16, 42 = 1
=
4
3
= 3
Step 3 Evaluate AC - B2 and try to classify the critical point 16, 42 by using
Theorem 2:
1 43 2 132
AC - B2 =
- 112 2 = 3 7 0
and
A =
4
3
7 0
Case 2 in Theorem 2 applies, and M1x, y2 has a local minimum at 16, 42.
If x = 6 and y = 4, then
48
48
z =
= 2
=
xy
162142
The dimensions that will require the least material are 6 inches by 4 inches
by 2 inches (see Fig. 6).
Matched Problem 4
If the box in Example 4 must have a volume of 384 cubic
inches, find the dimensions that will require the least material.
Exercises 7.3
W
Skills Warm-up Exercises
18. f1x, y2 = 15x2 - y 2 - 10y
In Problems 1–8, find f ′(0), f ″(0), and determine whether f has a
local minimum, local maximum, or neither at x = 0. (If necessary, review the second derivative test for local extrema in Section
4.5).
19. f1x, y2 = 8 - x2 + 12x - y 2 - 2y
1. f1x2 = 2x3 - 9x2 + 4
3. f1x2 =
1
1 - x2
4. f1x2 =
22. f1x, y2 = 4x2 - xy + y 2 + 12
23. f1x, y2 = 100 + 6xy - 4x2 - 3y 2
1
1 + x2
24. f1x, y2 = 5x2 - y 2 + 2y + 6
2
5. f1x2 = e-x
6. f1x2 = ex
25. f1x, y2 = x2 + xy + y 2 - 7x + 4y + 9
7. f1x2 = x3 - x2 + x - 1
8. f1x2 = 13x + 12 2
26. f1x, y2 = - x2 + 2xy - 2y 2 - 20x + 34y + 40
In Problems 9–16, find fx 1x, y2 and fy 1x, y2, and explain, using
Theorem 1, why f1x, y2 has no local extrema.
9. f1x, y2 = 4x + 5y - 6
10. f1x, y2 = 10 - 2x - 3y + x
2
3
11. f1x, y2 = 3.7 - 1.2x + 6.8y + 0.2y + x
4
12. f1x, y2 = x3 - y 2 + 7x + 3y + 1
13. f1x, y2 = - x2 + 2xy - y 2 - 4x + 5y
2
2
14. f1x, y2 = 3x - 12xy + 12y + 8x + 9y - 15
x
15. f1x, y2 = ye - 3x + 4y
16. f1x, y2 = y + y 2 ln x
B
21. f1x, y2 = x2 + 3xy + 2y 2 + 5
2. f1x2 = 4x3 + 6x2 + 100
2
A
20. f1x, y2 = x2 + y 2 + 6x - 8y + 10
In Problems 17–36, use Theorem 2 to find the local extrema.
17. f1x, y2 = x2 + 8x + y 2 + 25
C
27. f1x, y2 = exy
28. f1x, y2 = x2y - xy 2
29. f1x, y2 = x3 + y 3 - 3xy
30. f1x, y2 = 2y 3 - 6xy - x2
31. f1x, y2 = 2x4 + y 2 - 12xy
32. f1x, y2 = 16xy - x4 - 2y 2
33. f1x, y2 = x3 - 3xy 2 + 6y 2
34. f1x, y2 = 2x2 - 2x2y + 6y 3
35. f1x, y2 = xey + xy + 1
36. f1x, y2 = y ln x + 3xy
37. Explain why f1x, y2 = x2 has a local extremum at infinitely
many points.
SECTION 7.3 Maxima and Minima
38. (A) Find the local extrema of the functions
f1x, y2 = x + y, g1x, y2 = x2 + y 2, and
h1x, y2 = x3 + y 3.
(B) Discuss the local extrema of the function
k1x, y2 = xn + y n, where n is a positive integer.
39. (A) Show that (0, 0) is a critical point of the function
f1x, y2 = x4ey + x2y 4 + 1, but that the secondderivative test for local extrema fails.
(B) Use cross sections, as in Example 2, to decide whether
f has a local maximum, a local minimum, or a saddle
point at (0, 0).
40. (A) Show that (0, 0) is a critical point of the function
2
g1x, y2 = exy + x2y 3 + 2, but that the secondderivative test for local extrema fails.
(B) Use cross sections, as in Example 2, to decide whether
g has a local maximum, a local minimum, or a saddle
point at (0, 0).
Applications
41. Product mix for maximum profit. A firm produces two
types of earphones per year: x thousand of type A and y thousand of type B. If the revenue and cost equations for the year
are (in millions of dollars)
44. Maximizing profit. A store sells two brands of laptop sleeves.
The store pays $25 for each brand A sleeve and $30 for each
brand B sleeve. A consulting firm has estimated the daily demand equations for these two competitive products to be
x = 130 - 4p + q
Demand equation for brand A
y = 115 + 2p - 3q Demand equation for brand B
where p is the selling price for brand A and q is the selling
price for brand B.
(A) Determine the demands x and y when p = $40 and
q = $50; when p = $45 and q = $55.
(B) How should the store price each brand of sleeve to
maximize daily profits? What is the maximum daily
profit? [Hint: C = 25x + 30y, R = px + qy, and
P = R - C.]
45. Minimizing cost. A satellite TV station is to be located at
P1x, y2 so that the sum of the squares of the distances from
P to the three towns A, B, and C is a minimum (see figure).
Find the coordinates of P, the location that will minimize the
cost of providing satellite TV for all three towns.
y
10
B(2, 6)
5
P(x, y)
R1x, y2 = 2x + 3y
C1x, y2 = x2 - 2xy + 2y 2 + 6x - 9y + 5
determine how many of each type of earphone should be
produced per year to maximize profit. What is the maximum
profit?
42. Automation–labor mix for minimum cost. The annual
labor and automated equipment cost (in millions of dollars)
for a company’s production of HDTVs is given by
C1x, y2 = 2x2 + 2xy + 3y 2 - 16x - 18y + 54
where x is the amount spent per year on labor and y is the
amount spent per year on automated equipment (both in millions of dollars). Determine how much should be spent on each
per year to minimize this cost. What is the minimum cost?
43. Maximizing profit. A store sells two brands of camping
chairs. The store pays $60 for each brand A chair and $80 for
each brand B chair. The research department has estimated
the weekly demand equations for these two competitive
products to be
x = 260 - 3p + q Demand equation for brand A
y = 180 + p - 2q Demand equation for brand B
where p is the selling price for brand A and q is the selling
price for brand B.
(A) Determine the demands x and y when p = $100 and
q = $120; when p = $110 and q = $110.
(B) How should the store price each chair to maximize
weekly profits? What is the maximum weekly profit?
[Hint: C = 60x + 80y, R = px + qy, and P = R - C.]
507
A(0, 0)
5
C(10, 0)
x
46. Minimizing cost. Repeat Problem 45, replacing the coordinates of B with B16, 92 and the coordinates of C with C19, 02.
47. Minimum material. A rectangular box with no top and
two parallel partitions (see figure) must hold a volume of 64
cubic inches. Find the dimensions that will require the least
material.
48. Minimum material. A rectangular box with no top and two
intersecting partitions (see figure) must hold a volume of 72
cubic inches. Find the dimensions that will require the least
material.