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47. Repeat exercises 45 and 46 if the given graph is of (a) f or 57. A basic principle of physics is that light follows the path
′′
(b) f instead of f. of minimum time. Assuming that the speed of light in the
earth’s atmosphere decreases as altitude decreases, argue
48. Prove Theorem 5.2 (the Second Derivative Test). (Hint: that the path that light follows is concave down. Explain
′′
′′
Think about what the definition of f (c) says when f (c) > 0 why this means that the setting Sun appears higher in the
′′
or f (c) < 0.)
sky than it really is.
49. Show that the function in example 5.4 can be written as
2
f(x) = (x − 4) − 6. Conclude that the absolute minimum
2
of f is −6, occurring at x =±2. Do a similar analysis with
4
2
g(x) = x − 6x + 1.
4
3
2
50. For f(x) = x + bx + cx + dx + 2, show that there are two
3 2
inflection points if and only if c < b . Show that the sum
8
of the x-coordinates of the inflection points is −b∕2.
APPLICATIONS EXPLORATORY EXERCISES
51. Suppose that w(t) is the depth of water in a city’s water 1. The linear approximation that we defined in section 4.1 is
reservoir at time t. Which would be better news at time the line having the same location and the same slope as
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t = 0, w (0) = 0.05 or w (0) =−0.05, or would you need to the function being approximated. Since two points deter-
′
know the value of w (0) to determine which is better? mine a line, two requirements (point, slope) are all that a
52. Suppose that T(t) is a sick person’s temperature at time linear function can satisfy. However, a quadratic function
′′
t. Which would be better news at time t, T (0) = 2 or can satisfy three requirements, since three points deter-
′
′′
T (0) =−2, or would you need to know the value of T (0) mine a parabola (and there are three constants in a gen-
2
and T(0) to determine which is better? eral quadratic function ax + bx + c). Suppose we want to
define a quadratic approximation to f(x) at x = a. Build-
53. Suppose that a company that spends AEDx thousand on ing on the linear approximation, the general form is g(x) =
advertising sells AED s(x) of merchandise, where s(x) = f(a) + f (a)(x − a) + c(x − a) for some constant c to be de-
′
2
3
2
−3x + 270x − 3600x + 18,000. Find the value of x that ma- termined. In this way, show that g(a) = f(a) and g (a) =
′
ximize the rate of change of sales. (Hint: Read the ques- f (a). That is, g(x) has the right position and slope at
′
tion carefully!) Find the inflection point and explain why in x = a. The third requirement is that g(x) have the right
advertising terms this is the “point of diminishing returns.” concavity at x = a, so that g (a) = f (a). Find the constant c
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54. The number of units Q that a worker has produced in a day that makes this true. Then, find such a quadratic approxi-
x
is related to the number of hours t since the work day be- mation for each of the functions sin x, cos x and e at x = 0.
3
2
′
gan. Suppose that Q(t) =−t + 6t + 12t. Explain why Q (t) In each case, graph the original function, linear approxima-
is a measure of the efficiency of the worker at time t. Find tion and quadratic approximation, and describe how close
the time at which the worker’s efficiency is a maximum. the approximations are to the original functions.
Explain why it is reasonable to call the inflection point the 2. In this exercise, we explore a basic problem in genetics.
“point of diminishing returns.” Suppose that a species reproduces according to the follow-
55. Suppose that it costs a company C(x) = 0.01x + 40x + 3600 ing probabilities: p is the probability of having no chil-
2
0
dirhams to manufacture x units of a product. For this cost dren, p is the probability of having one offspring, p is the
2
1
C(x) probability of having two offspring, ..., p is the probabil-
n
function, the average cost function is C(x) = x . Find the ity of having n offspring and n is the largest number of off-
value of x that minimizes the average cost. The cost function spring possible. Explain why for each i,wehave0 ≤ p ≤ 1
i
Galyna Andrushko/Shutterstock.com 56. The antiarrhythmic medicine lidocaine slowly decays after tive solution of the equation F(x) = x for 0 ≤ x ≤ 1 rep-
and p + p + p + ⋅⋅⋅ + p = 1. We define the function
can be related to the efficiency of the production process.
1
0
2
n
F(x) = p + p x + p x + ⋅⋅⋅ + p x . The smallest nonnega-
2
n
Explain why a cost function that is concave down indicates
2
n
0
1
better efficiency than a cost function that is concave up.
resents the probability that the species becomes extinct.
′
Show graphically that if p > 0 and F (1) > 1, then there is
0
entering the bloodstream. The plasma concentration t min-
a solution of F(x) = x with 0 < x < 1. Thus, there is a posi-
utes after administering the drug can be modeled by
′
tive probability of survival. However, if p > 0 and F (1) < 1,
0
show that there are no solutions of F(x) = x with 0 < x < 1.
−t∕65.7
)
−t∕13.3
−t∕6.55
(
c(t) = 92.8 −0.129e
+ 0.218e
− 0.089e
.
(Hint: First show that F is increasing and concave up.)
Copyright © McGraw-Hill Education Use a CAS to estimate the time of maximum concentration 3. Give as complete a description of the graph of f(x) = x − 1
x + c
and inflection point (t > 0). Suppose that f(t) represents the
2
as possible. In particular, find the values of c for which there
concentration of another type of medicine. If the graph of
are two critical points (or one critical point, or no critical
f(t) has a similar shape and the same maximum point as
points) and identify any extrema. Similarly, determine how
the graph of c(t) but the inflection point occurs at a larger
value of t, would this medicine be more or less effective than
the existence or not of inflection points depends on the
value of c.
lidocaine? Briefly explain.
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