SECTION 7.3 word.docx

SECTION 7.3 Optimizing Functions of Two Variables
Relative Extrema ■ The function f (x, y) is said to have a relative maximum
at the point P(a, b) in the domain of f if f (a, b) ≥ f (x, y) for all points (x, y) in a
circular disk centered at P. Similarly, if f (c, d ) ≤ f (x, y) for all points (x, y) in a
circular disk centered at Q, then f(x, y) has a relative minimum at Q(c, d).
588 CHAPTER 7 Calculus of Several Variables 7-32
79. PACKAGING A soft drink can is H centimeters
(cm) tall and has a radius of R cm. The cost of
material in the can is 0.0005 cents per cm2
and
the soda itself costs 0.001 cents per cm3
.
a. Find a function C(R, H) for the cost of the mate-
rials and contents of a can of soda. (You will
need the formulas for volume and surface area
given in Exercises 53 and 54.)
b. The cans are currently 12 cm tall and have a
radius of 3 cm. Use calculus to estimate the ef-
fect on cost of increasing the radius by 0.3 cm
and decreasing the height by 0.2 cm.
80. SLOPE OF A LEVEL CURVE Suppose
y = h(x) is a differentiable function of x and that
f(x, y) = C for some constant C. Use the chain
rule (with x taking the role of t) to show that
=f =f dy
+ = 0
=x =y dx
Conclude that the slope at each point (x, y) on the
level curve F(x, y) = C is given by
dy
= —
fx
dx fy
81. Use the formula obtained in Exercise 80 to find
the slope of the level curve
x2
+ xy + y3
= 1
at the point (—1, 1). What is the equation of the
tangent line to the level curve at this point?
82. Use the formula obtained in Exercise 80 to find
the slope of the level curve
x2
y + 2y3
— 2e—x
= 14
at the point (0, 2). What is the equation of the
tangent line to the level curve at this point?
83. INVESTMENT SATISFACTION Suppose a
particular investor derives U(x, y) units of
satisfaction from owning x stock units and y bond
units, where
U(x, y) = (2x + 3)(y + 5)
The investor currently owns x = 27 stock units
and y = 12 bond units.
a. Find the marginal utilities Ux and Uy.
b. Evaluate Ux and Uy for the current values of x
and y.
c. Use calculus to estimate how the investor’s sat-
isfaction changes if she adds 3 stock units and
removes 2 bond units from her portfolio.
d. Estimate how many bond units the investor
could substitute for 1 stock unit without affect-
ing her total satisfaction with her portfolio.
Suppose a manufacturer produces two DVD player models, the deluxe and the stan-
dard, and that the total cost of producing x units of the deluxe and y units of the
standard is given by the function C(x, y). How would you find the level of production
x = a and y = b that results in minimal cost? Or perhaps the output of a certain pro-
duction process is given by Q(K, L), where K and L measure capital and labor expen-
diture, respectively. What levels of expenditure K0 and L0 result in maximum output?
In Section 3.4, you learned how to use the derivative f '(x) to find the largest and
smallest values of a function of a single variable f(x), and the goal of this section is to
extend those methods to functions of two variables f(x, y). We begin with a definition.
In geometric terms, there is a relative maximum of f(x, y) at P(a, b) if the sur-
face z = f(x, y) has a “peak” at the point (a, b, f(a, b)); that is, if (a, b, f(a, b)) is at

z
Relative maximum
(a, b, f (a, b)) Surface z = f (x, y)
(c, d, f (c, d)) Relative minimum
y
P(a, b)
Q(c, d)
x
z z
Horizontal
tangent
(a, b, f(a, b))
(a, b, f(a, b))
Horizontal
tangent
y y
(a, b)
(a, b)
x x
(a) For y = b, the slope fx(a, b) = 0 (b) For x = a, the slope fy(a, b) = 0
7-33 SECTION 7.3 OPTIMIZING FUNCTIONS OF TWO VARIABLES 589
least as high as any nearby point on the surface. Similarly, a relative minimum of
f (x, y) occurs at Q(c, d ) if the point (c, d, f (c, d)) is at the bottom of a “valley,” so
(c, d, f (c, d )) is at least as low as any nearby point on the surface. For example, in
Figure 7.11, the function f (x, y) has a relative maximum at P(a, b) and a relative min-
imum at Q(c, d).
FIGURE 7.11 Relative extrema of the function f(x, y).
Critical Points The points (a, b) in the domain of f(x, y) for which both fx(a, b) = 0 and fy(a, b) = 0
are said to be critical points of f. Like the critical numbers for functions of one variable,
these critical points play an important role in the study of relative maxima and minima.
To see the connection between critical points and relative extrema, suppose f(x, y)
has a relative maximum at (a, b). Then the curve formed by intersecting the surface
z = f (x, y) with the vertical plane y = b has a relative maximum and hence a
horizontal tangent line when x = a (Figure 7.12a). Since the partial derivative fx(a, b)
is the slope of this tangent line, it follows that fx(a, b) = 0. Similarly, the curve formed
by intersecting the surface z = f (x, y) with the plane x = a has a relative maximum
when y = b (Figure 7.12b), and so fy(a, b) = 0. This shows that a point at which
a function of two variables has a relative maximum must be a critical point. A similar
argument shows that a point at which a function of two variables has a relative
minimum must also be a critical point.
FIGURE 7.12 The partial derivatives are zero at a relative extremum.

z
Saddle point
y
x
Here is a more precise statement of the situation.
Saddle Points
FIGURE 7.13 The saddle
surface z = y2
— x2
.
The Second
Partials Test
Although all the relative extrema of a function must occur at critical points, not every
critical point of a function corresponds to a relative extremum. For example, if
f(x, y) = y2
— x2
, then
fx(x, y) = —2x and fy(x, y) = 2y
so fx(0, 0) = fy(0, 0) = 0. Thus, the origin (0, 0) is a critical point for f (x, y), and the
surface z = y2
— x2
has horizontal tangents at the origin along both the x axis and the
y axis. However, in the xz plane (where y = 0) the surface has the equation z = —x2
,
which is a downward opening parabola, while in the yz plane (where x = 0), we
have the upward opening parabola z = y2. This means that at the origin, the surface
z = y2
— x2
has a relative maximum in the “x direction” and a relative minimum in
the “y direction.”
Instead of having a “peak” or a “valley” above the critical point (0, 0), the surface
z = y2 — x2 is shaped like a “saddle,” as shown in Figure 7.13, and for this reason is
called a saddle surface. For a critical point to correspond to a relative extremum, the
same extreme behavior (maximum or minimum) must occur in all directions. Any
critical point (like the origin in this example) where there is a relative maximum in
one direction and a relative minimum in another direction is called a saddle point.
Here is a procedure involving second-order partial derivatives that you can use to
decide whether a given critical point is a relative maximum, a relative minimum, or
a saddle point. This procedure is the two-variable version of the second derivative
test for functions of a single variable that you saw in Section 3.2.
The Second Partials Test
Let f(x, y) be a function of x and y whose partial derivatives fx, fy, fxx, fyy, and fxy
all exist, and let D(x, y) be the function
D(x, y) = fxx(x, y) fyy(x, y) — [ fxy(x, y)]2
Step 1. Find all critical points of f(x, y); that is, all points (a, b) so that
fx(a, b) = 0 and fy(a, b) = 0
Step 2. For each critical point (a, b) found in step 1, evaluate D(a, b).
Step 3. If D(a, b) < 0, there is a saddle point at (a, b).
Step 4. If D(a, b) > 0, compute fxx(a, b):
If fxx(a, b) > 0, there is a relative minimum at (a, b).
If fxx(a, b) < 0, there is a relative maximum at (a, b).
If D(a, b) = 0, the test is inconclusive and f may have either a relative extremum
or a saddle point at (a, b).
Critical Points and Relative Extrema ■ A point (a, b) in the domain of
f (x, y) for which the partial derivatives fx and fy both exist is called a critical point
of f if both
fx(a, b) = 0 and fy(a, b) = 0
If the ﬁrst-order partial derivatives of f exist at all points in some region R in
the xy plane, then the relative extrema of f in R can occur only at critical points.

EXAMPLE 7.3.1
Notice that there is a saddle point at the critical point (a, b) only when the
quantity D in the second partials test is negative. If D is positive, there is either a
relative maximum or a relative minimum in all directions. To decide which, you
can restrict your attention to any one direction (say, the x direction) and use the
sign of the second partial derivative fxx in exactly the same way as the single vari-
able second derivative was used in the second derivative test given in Chapter 3;
namely,
a relative minimum if fxx(a, b) > 0
a relative maximum if fxx(a, b) < 0
You may ﬁnd the following tabular summary a convenient way of remembering the
conclusions of the second partials test:
Sign of D Sign of fxx Behavior at (a, b)
+
+
—
+
—
Relative minimum
Relative maximum
Saddle point
The proof of the second partials test involves ideas beyond the scope of this
text and is omitted. Examples 7.3.1 through 7.3.3 illustrate how the test can be
used.
Find all critical points for the function f(x, y) = x2
+ y2
and classify each as a rela-
tive maximum, a relative minimum, or a saddle point.
Solution
Since
fx = 2x and fy = 2y
the only critical point of f is (0, 0). To test this point, use the second-order partial
derivatives
to get
fxx = 2 fyy = 2 and fxy = 0
D(x, y) = fxx fyy — ( fxy)2
= (2)(2) — 02
= 4
That is, D(x, y) = 4 for all points (x, y) and, in particular,
D(0, 0) = 4 > 0
Hence, f has a relative extremum at (0, 0). Moreover, since
fxx(0, 0) = 2 > 0
it follows that the relative extremum at (0, 0) is a relative minimum. For reference,
the graph of f is sketched in Figure 7.14.
EXPLORE!
Refer to Example 7.3.1. Store
f(x, y) = x2
+ y2
in the equation
editor as Y1 = X2
+ L12
, where
L1 = {—1, —0.6, 0, 0.8, 1.2}.
Graph using the window [—
3, 3]1 by [—1, 5]1 and the
graphing style showing a ball
with trailer. Pay close attention
to the order of the curves as
they represent cross sections
of the function at the speciﬁc y
values listed in L1. Describe
what you observe.

z
y
Relative minimum
x
EXAMPLE 7.3.2
FIGURE 7.14 The surface z = x2
+ y2
with a relative minimum at (0, 0).
Find all critical points for the function f(x, y) = 12x — x3
— 4y2
and classify each as
a relative maximum, a relative minimum, or a saddle point.
Solution
Since
fx = 12 — 3x2
and fy = —8y
you find the critical points by solving simultaneously the two equations
12 — 3x2
= 0
—8y = 0
From the second equation, you get y = 0 and from the first,
3x2
= 12
x = 2 or —2
Thus, there are two critical points, (2, 0) and (—2, 0).
To determine the nature of these points, you first compute
fxx = —6x fyy = —8 and fxy = 0
and then form the function
D = fxx fyy — ( fxy)2 = (—6x)(—8) — 0 = 48x
Applying the second partials test to the two critical points, you find
D(2, 0) = 48(2) = 96 > 0 and fxx(2, 0) = —6(2) = —12 < 0
and
D(—2, 0) = 48(—2) = —96 < 0

2
( )
EXAMPLE 7.3.3
so a relative maximum occurs at (2, 0) and a saddle point at (—2, 0). These results
are summarized in this table.
Critical point (a, b) Sign of D(a, b) Sign of fxx(a, b) Behavior at (a, b)
(2, 0) + — Relative maximum
(—2, 0) — Saddle point
Solving the equations fx = 0 and fy = 0 simultaneously to find the critical points
of a function of two variables is rarely as simple as in Examples 7.3.1 and 7.3.2. The
algebra in Example 7.3.3 is more typical. Before proceeding, you may wish to refer
to Appendix A2, in which techniques for solving systems of two equations in two
unknowns are discussed.
Find all critical points for the function f(x, y) = x3
— y3
+ 6xy and classify each as a
relative maximum, a relative minimum, or a saddle point.
Solution
Since
Just-In-Time REVIEW
fx = 3x2
+ 6y and fy = —3y2
+ 6x
you find the critical points of f by solving simultaneously the two equations
3x2
+ 6y = 0 and —3y2
+ 6x = 0
x2
From the first equation, you get y = —
2
which you can substitute into the second
Recall that
a3
— b3
= (a — b)(a2
+ ab + b2
)
so that
x3
— 8 = (x — 2)(x2
+ 2x + 4).
2
equation to find
—x2 2
—3
3x4
+ 6x = 0
Because x + 2x + 4 = 0 has
—
4
+ 6x = 0
no real solutions (as can be
seen using the quadratic
equation), the only real
solution of x3
— 8 = 0 is
x = 2.
—x(x3
— 8) = 0
The solutions of this equation are x = 0 and x = 2. These are the x coordinates of the
critical points of f. To get the corresponding y coordinates, substitute these values of
x2
x into the equation y = —
2
(or into either one of the two original equations). You
will find that y = 0 when x = 0 and y = —2 when x = 2. It follows that the critical
points of f are (0, 0) and (2, —2).
The second-order partial derivatives of f are
fxx = 6x fyy = —6y and fxy = 6
Hence,
D(x, y) = fxx fyy — ( fxy)2 = —36xy — 36 = —36(xy + 1)

profit of the local brand of the national brand
Since
D(0, 0) = —36[(0)(0) + 1] = —36 < 0
it follows that f has a saddle point at (0, 0). Since
D(2, —2) = —36[2(—2) + 1] = 108 > 0
and
fxx(2, —2) = 6(2) = 12 > 0
you see that f has a relative minimum at (2, —2). To summarize:
Practical
Optimization
Problems
Critical point (a, b) D(a, b) fxx(a, b) Behavior at (a, b)
(0, 0) — Saddle point
(2, —2) + + Relative minimum
In Example 7.3.4, you will use the theory of relative extrema to solve an opti-
mization problem from economics. Actually, you will be trying to find the absolute
maximum of a certain function, which turns out to coincide with the relative max-
imum of the function. This is typical of two-variable optimization problems in the
social and life sciences, and in this text, you can assume that a relative extremum
you find as the solution to any practical optimization problem is actually the
absolute extremum.
A grocery store carries two brands of cat food, a local brand that it obtains at the cost
of 30 cents per can and a well-known national brand it obtains at the cost of 40 cents
per can. The grocer estimates that if the local brand is sold for x cents per can and the
national brand for y cents per can, then approximately 70 — 5x + 4y cans of the local
brand and 80 + 6x — 7y cans of the national brand will be sold each day. How should
the grocer price each brand to maximize total daily profit from the sale of cat food?
Solution
Since
(Total
)= (profit from the sale
)+ (profit from the sale
)
it follows that the total daily profit from the sale of the cat food is given by the
function
f(x, y) = (70 — 5x + 4y) · (x — 30) + (80 + 6x — 7y) · (y — 40)
aeeeeeebeeeeeec aeebeec aeeeeeebeeeeeec aeebeec
items sold profit per item items sold profit per item
local brand national brand
= —5x2
+ 10xy — 20x — 7y2
+ 240y — 5,300
Compute the partial derivatives
fx = —10x + 10y — 20 and fy = 10x — 14y + 240
EXAMPLE 7.3.4

xy
EXAMPLE 7.3.5
and set them equal to zero to get
—10x + 10y — 20 = 0 and 10x — 14y + 240 = 0
or
—x + y = 2 and 5x — 7y = —120
Then solve these equations simultaneously to get
x = 53 and y = 55
It follows that (53, 55) is the only critical point of f.
Next apply the second partials test. Since
fxx = —10 fyy = —14 and fxy = 10
you get
D(x, y) = fxx fyy — ( fxy)2
= (—10)(—14) — (10)2
= 40
Because you have
D(53, 55) = 40 > 0 and fxx(53, 55) = —10 < 0
it follows that f has a (relative) maximum when x = 53 and y = 55. That is, the gro-
cer can maximize proﬁt by selling the local brand of cat food for 53 cents per can
and the national brand for 55 cents per can.
FIGURE 7.15 Locations of
The business manager for Acme Corporation plots a grid on a map of the region Acme
serves and determines that the company’s three most important customers are located
at points A(1, 5), B(0, 0), and C(8, 0), where units are in miles. At what point W(x, y)
should a warehouse be located in order to minimize the sum of the squares of the dis-
tances from W to A, B, and C (Figure 7.15)?
Solution
The sum of the squares of the distances from W to A, B, and C is given by the function
S(x, y) = [(x — 1)2
+ (y — 5)2
] + (x2
+ y2
) + [(x — 8)2
+ y2
]
businesses A, B, and C and aeebeec aeeeeeeeeebeeeeeeeeec aeebeec aeeeeeebeeeeeec
warehouse W.
sum of squares square of distance square of distance square of distance
of distances from W to A from W to B from W to C
To minimize S(x, y), you begin by computing the partial derivatives
Sx = 2(x — 1) + 2x + 2(x — 8) = 6x — 18
Sy = 2(y — 5) + 2y + 2y = 6y — 10
Then Sx = 0 and Sy = 0 when
6x — 18 = 0
6y — 10 = 0
or x = 3 and y =
5
. Since S = 6, S = 0, and S = 6, you get
3 xx xy yy
D = SxxSyy — S2
= (6)(6) — 02
= 36 > 0
y
A(1, 5)
W(x, y)
B(0, 0) C(8, 0)
x

3
3
x
and
Sxx(3,
5
)= 6 > 0
Thus, the sum of squares is minimized at the map point W(3,
5
).
EXERCISES ■ 7.3
In Exercises 1 through 22, find the critical points of the
given functions and classify each as a relative
maximum, a relative minimum, or a saddle point.
(Note: The algebra in Exercises 19 through 22 is
challenging.)
1. f(x, y) = 5 — x2 — y2
2. f(x, y) = 2x2 — 3y2
3. f(x, y) = xy
4. f(x, y) = x2
+ 2y2
— xy + 14y
5. f(x, y) =
16
+
6
+ x2
— 3y2
x y
6. f(x, y) = xy +
8
+
8
23. RETAIL SALES A T-shirt shop carries two
competing shirts, one endorsed by Tim Duncan
and the other by LeBron James. The owner of the
store can obtain both types at a cost of $2 per
shirt and estimates that if Duncan shirts are sold
for x dollars apiece and James shirts for y dollars
apiece, consumers will buy 40 — 50x + 40y
Duncan shirts and 20 + 60x — 70y James shirts
each day. How should the owner price the shirts
in order to generate the largest possible profit?
24. PRICING The telephone company is planning
to introduce two new types of executive
communications systems that it hopes to sell to
its largest commercial customers. It is estimated
x y that if the first type of system is priced at
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
f(x, y) = 2x3 + y3 + 3x2 — 3y — 12x — 4
f(x, y) = (x — 1)2
+ y3
— 3y2
— 9y + 5
f(x, y) = x3
+ y2
— 6xy + 9x + 5y + 2
f(x, y) = —x4
— 32x + y3
— 12y + 7
f(x, y) = xy2 — 6x2 — 3y2
f(x, y) = x2
— 6xy — 2y3
f(x, y) = (x2
+ 2y2
)e1—x
2
—y2
f(x, y) = e—(x2
+y2
—6y)
f (x, y) = x3
— 4xy + y3
f(x, y) = (x — 4) ln (xy)
f(x, y) = 4xy — 2x4
— y2
+ 4x — 2y
f(x, y) = 2x4 + x2 + 2xy + 3x + y2 + 2y + 5
f(x, y) =
1
x2
+ y2
+ 3x — 2y + 1
f(x, y) = xye—(16x2
+9y2
)/288
(y2
)
x hundred dollars per system and the second type
at y hundred dollars per system, approximately
40 — 8x + 5y consumers will buy the first type
and 50 + 9x — 7y will buy the second type. If the
cost of manufacturing the first type is $1,000 per
system and the cost of manufacturing the second
type is $3,000 per system, how should the
telephone company price the systems to generate
the largest possible profit?
25. CONSTRUCTION Suppose you wish to
construct a rectangular box with a volume of
32 ft3
. Three different materials will be used in
the construction. The material for the sides costs
$1 per square foot, the material for the bottom
costs $3 per square foot, and the material for the
top costs $5 per square foot. What are the
dimensions of the least expensive such box?
26. CONSTRUCTION A farmer wishes to fence off
a rectangular pasture along the bank of a river.
The area of the pasture is to be 6,400 yd2
, and no
22. f(x, y) =
x
x2
+ y2
+ 4
fencing is needed along the river bank. Find the
21. f(x, y) = x ln + 3x — xy2

( )
dimensions of the pasture that will require the
least amount of fencing.
27. RETAIL SALES A company produces x units
of commodity A and y units of commodity B.
All the units can be sold for p = 100 — x
dollars per unit of A and q = 100 — y dollars
per unit of B. The cost (in dollars) of producing
these units is given by the joint-cost function
C(x, y) = x2 + xy + y2. What should x and y be
to maximize profit?
28. RETAIL SALES Repeat Exercise 27 for the case
where p = 20 — 5x, q = 4 — 2y, and C = 2xy + 4.
29. RESPONSE TO STIMULI Consider an
experiment in which a subject performs a task
while being exposed to two different stimuli (for
example, sound and light). For low levels of the
stimuli, the subject’s performance might actually
improve, but as the stimuli increase, they
eventually become a distraction and the
performance begins to deteriorate. Suppose in a
certain experiment in which x units of stimulus A
and y units of stimulus B are applied, the
performance of a subject is measured by the
function
f(x, y) = C + xye1—x
2
—y2
b. The function given in part (a) is artificial, but
the ideas are not. Research the topic of ethics in
industry and write a paragraph on how you feel
these choices should be made.*
31. PARTICLE PHYSICS A particle of mass m in a
rectangular box with dimensions x, y, and z has
ground state energy
k2 1 1 1
E(x, y, z) =
8m x2 +
y2 +
z2
where k is a physical constant. If the volume of
the box satisfies xyz = V0 for constant V0, find the
values of x, y, and z that minimize the ground
state energy.
32. ALLOCATION OF FUNDS A manufacturer is
planning to sell a new product at the price of $210
per unit and estimates that if x thousand dollars is
spent on development and y thousand dollars is
spent on promotion, consumers will buy
approximately
640y
+
216x
units of the product.
y + 3 x + 5
If manufacturing costs for this product are $135
per unit, how much should the manufacturer
spend on development and how much on
promotion to generate the largest possible profit
from the sale of this product? [Hint: Profit
where C is a positive constant. How many units
of each stimulus result in maximum performance?
30. SOCIAL CHOICES The social desirability of an
enterprise often involves making a choice between
the commercial advantage of the enterprise and
the social or ecological loss that may result. For
instance, the lumber industry provides paper
products to society and income to many workers
and entrepreneurs, but the gain may be offset by
the destruction of habitable territory for spotted
owls and other endangered species. Suppose the
social desirability of a particular enterprise is
measured by the function
D(x, y) = (16 — 6x)x — (y2
— 4xy + 40)
where x measures commercial advantage (profit
and jobs) and y measures ecological disadvantage
(species displacement, as a percentage) with
x ≥ 0 and y ≥ 0. The enterprise is deemed
desirable if D ≥ 0 and undesirable if D < 0.
a. What values of x and y will maximize social de-
sirability? Interpret your result. Is it possible for
this enterprise to be desirable?
=
(number of units)(price per unit — cost per unit) —
total amount spent on development and promotion.]
33. PROFIT UNDER MONOPOLY A manufac-
turer with exclusive rights to a sophisticated new
industrial machine is planning to sell a limited
number of the machines to both foreign and
domestic firms. The price the manufacturer can
expect to receive for the machines will depend
on the number of machines made available. (For
example, if only a few of the machines are
placed on the market, competitive bidding among
prospective purchasers will tend to drive the price
up.) It is estimated that if the manufacturer
supplies x machines to the domestic market
and y machines to the foreign market, the
machines will sell for 60 —
x
+
y
thousand dol-
5 20
y x
lars apiece domestically and for 50 —
10
+
20
thousand dollars apiece abroad. If the manufacturer
*Start with the article by K. R. Stollery, “Environmental Controls in
Extractive Industries,” Land Economics, Vol. 61, 1985, p. 169.

can produce the machines at the cost of $10,000
apiece, how many should be supplied to each
market to generate the largest possible profit?
34. PROFIT UNDER MONOPOLY A
manufacturer with exclusive rights to a new
industrial machine is planning to sell a limited
number of them and estimates that if x machines
are supplied to the domestic market and y to the
foreign market, the machines will sell for 150 —
x
6
thousand dollars apiece domestically and for
minimize the sum of squares of the distances from
the towns?
36. MAINTENANCE In relation to a rectangular
map grid, four oil rigs are located at the points
(—300, 0), (—100, 500), (0, 0), and (400, 300)
where units are in feet. Where should a
maintenance shed M(a, b) be located to minimize
the sum of squares of the distances from the rigs?
37. GENETICS Alternative forms of a gene are
called alleles. Three alleles, designated A, B, and
O, determine the four human blood types A, B, O,
100 —
y
20
thousand dollars apiece abroad.
and AB. Suppose that p, q, and r are the pro-
portions of A, B, and O in a particular population,
a. How many machines should the manufacturer
supply to the domestic market to generate the
largest possible profit at home?
b. How many machines should the manufacturer
supply to the foreign market to generate the
largest possible profit abroad?
c. How many machines should the manufacturer
supply to each market to generate the largest
possible total profit?
d. Is the relationship between the answers in parts
(a), (b), and (c) accidental? Explain. Does a sim-
ilar relationship hold in Exercise 33? What ac-
counts for the difference between these two
problems in this respect?
35. CITY PLANNING Four small towns in a rural
area wish to pool their resources to build a
television station. If the towns are located at the
points (—5, 0), (1, 7), (9, 0), and (0, —8) on a
rectangular map grid, where units are in miles, at
what point S(a, b) should the station be located to
so that p + q + r = 1. Then according to the
Hardy-Weinberg law in genetics, the proportion
of individuals in the population who carry two
different alleles is given by P = 2pq + 2pr + 2rq.
What is the largest value of P?
38. LEARNING In a learning experiment, a subject
is first given x minutes to examine a list of facts.
The fact sheet is then taken away and the subject
is allowed y minutes to prepare mentally for an
exam based on the fact sheet. Suppose it is found
that the score achieved by a particular subject is
related to x and y by the formula
S(x, y) = —x2
+ xy + 10x — y2
+ y + 15
a. What score does the subject achieve if he takes
the test “cold” (with no study or contemplation)?
b. How much time should the subject spend in
study and contemplation to maximize his score?
What is the maximum score?
39. Tom, Dick, and Mary are participating in a cross-country relay race. Tom will
trudge as fast as he can through thick woods to the edge of a river; then Dick
will take over and row to the opposite shore. Finally, Mary will take the baton
and run along the river road to the finish line. The course is shown in the
accompanying figure. Teams must start at point S and finish at point F, but
they may position one member anywhere along the shore of the river and
another anywhere along the river road.
a. Suppose Tom can trudge at 2 mph, Dick can row at 4 mph, and Mary can run
at 6 mph. Where should Dick and Mary wait to receive the baton in order for
the team to finish the course as quickly as possible?
b. The main competition for Tom, Dick, and Mary is the team of Ann, Jan, and
Phineas. If Ann can trudge at 1.7 mph, Jan can row at 3.5 mph, and Phineas
can run at 6.3 mph, which team should win? By how much?
c. Does this exercise remind you of the spy story in Exercise 19 of Section 3.5? Cre-
ate your own spy story problem based on the mathematical ideas in this exercise.

2.5 mi
1.2 mi
S(0, 0)
EXERCISE 39
EXERCISE 40
40. CANCER THERAPY Certain malignant tumors that do not respond to
conventional methods of treatment such as surgery or chemotherapy may be
treated by hyperthermia, which involves applying extreme heat to tumors
using microwave transmissions.* One particular kind of microwave applicator
used in this therapy produces an absorbed energy density that falls off
exponentially. Specifically, the temperature at each point located r units from
the central axis of a tumor and z units inside the tumor is given by a formula
of the form
T(r, z) = Ae—pr
2
(e—qz
— e—sz
)
where A, p, q, and s are positive constants that depend on properties of
both blood and the heating appliance. At what depth inside the tumor
does maximum temperature occur? Express your answer in terms of A, p, q,
and r.
41. LIVABLE SPACE Define the livable space of a building to be the volume
of space in the building where a person 6 feet tall can walk (upright). An A-
frame cabin is y feet long and has equilateral triangular ends x feet on a side,
as shown in the accompanying figure. If the surface area of the cabin (roof
and two ends) is to be 500 ft2
, what dimensions x and y will maximize the
livable space?
Livable
space
6 ft
x
EXERCISE 41
*The ideas in this essay are based on the article by Leah Edelstein-Keshet, “Heat Therapy for Tumors,”
UMAP Modules 1991: Tools for Teaching, Lexington, MA: Consortium for Mathematics and Its
Applications, Inc., 1992, pp. 73–101.
x
x
x y
Woods
Tom
Dick y
x
River
F(4.3, 3.7)
Mary
4.3 mi
Road
Heat
applicator
Coolant
Skin r
Tumor
z

4t
y y
f = –3 f = 3
f = 1
f = –2 f = 2
f = 1
2 f = –1 f = 1
x x
f = 1 f = 3 f = –3
3
f = 2 f = –2
f = 1 f = –1
(a) (b)
42. BUTTERFLY WING PATTERNS The beautiful
patterns on the wings of butterflies have long
been a subject of curiosity and scientific study.
Mathematical models used to study these patterns
often focus on determining the level of morphogen
(a chemical that effects change). In a model
dealing with eyespot patterns,* a quantity of
morphogen is released from an eyespot and the
morphogen concentration t days later is given by
c. It turns out that M(z) is what is really needed to
analyze the eyespot wing pattern. Read pages
461–468 in the text cited with this problem, and
write a paragraph on how biology and mathe-
matics are blended in the study of butterfly wing
patterns.
43. Let f(x, y) = x2
+ y2
— 4xy. Show that f does not
have a relative minimum at its critical point
(0, 0), even though it does have a relative
S(r, t) =
1
e
√4πt
—(çkt+
r2
) t > 0
minimum at (0, 0) in both the x and y directions.
[Hint: Consider the direction defined by the line
where r measures the radius of the region on the
wing affected by the morphogen, and k and ç are
positive constants.
a. Find t so that
=S
= 0. Show that the function
m
=t
Sm(t) formed from S(r, t) by fixing r has a relative
y = x. That is, substitute x for y in the formula for
f and analyze the resulting function of x.]
In Exercises 44 through 47, find the partial derivatives
fx and fy and then use your graphing utility to
determine the critical points of each function.
maximum at tm. Is this the same as saying that the
function of two variables S(r, t) has a relative
maximum?
44.
45.
f(x, y) = (x2
+ 3y — 5)e—x
2
—2y2
x2 + xy + 7y2
f(x, y) =
x ln y
b. Let M(r) denote the maximum found in part (a);
that is, M(r) = S(r, tm). Find an expression for
M in terms of z = (1 + 4çkr2
)1/2
.
46. f(x, y) = 6x2
+ 12xy + y4
+ x — 16y — 3
47. f(x, y) = 2x4
+ y4
— x2
(11y — 18)
48. LEVEL CURVES Sometimes you can classify the critical points of a
function by inspecting its level curves. In each case shown in the
accompanying figure, determine the nature of the critical point of f at
(0, 0).
EXERCISE 48
*J. D. Murray, Mathematical Biology, 2nd ed., New York: Springer-Verlag, 1993, pp. 461–468.

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