3.3. GLOBAL OPTIMIZATION
183
in Figure 3.17, we see a function f that has a global maximum at x = c and a relative
maximum at x = a, since f (c) is greater than f (x) for every value of x, while f (a) is only
greater than the value of f (x) for x near a. Since the function appears to decrease without
bound, f has no global minimum, though clearly f has a relative minimum at x = b.
Our emphasis in this section is on finding the global extreme values of a function (if
they exist). In so doing, we will either be interested in the behavior of the function over its
entire domain or on some restricted portion. The former situation is familiar and similar
to work that we did in the two preceding sections of the text. We explore this through a
particular example in the following preview activity.
Preview Activity 3.3. Let f (x) = 2 +
3
1 + (x + 1) 2 .
(a) Determine all of the critical numbers of f .
(b) Construct a first derivative sign chart for f and thus determine all intervals on
which f is increasing or decreasing.
(c) Does f have a global maximum? If so, why, and what is its value and where is the
maximum attained? If not, explain why.
(d) Determine lim
x→∞
f (x) and lim
x→−∞
f (x).
(e) Explain why f (x) > 2 for every value of x.
(f) Does f have a global minimum? If so, why, and what is its value and where is the
minimum attained? If not, explain why.
⊲⊳
Global Optimization
For the functions in Figure 3.17 and Preview Activity 3.3, we were interested in finding
the global minimum and global maximum on the entire domain, which turned out to be
(−∞, ∞) for each. At other times, our perspective on a function might be more focused due
to some restriction on its domain. For example, rather than considering f (x) = 2 +
3
1+(x+1) 2
for every value of x, perhaps instead we are only interested in those x for which 0 ≤ x ≤ 4,
and we would like to know which values of x in the interval [0, 4] produce the largest
possible and smallest possible values of f . We are accustomed to critical numbers playing
a key role in determining the location of extreme values of a function; now, by restricting
the domain to an interval, it makes sense that the endpoints of the interval will also be
important to consider, as we see in the following activity. When limiting ourselves to a
particular interval, we will often refer to the absolute maximum or minimum value, rather
than the global maximum or minimum.
183
in Figure 3.17, we see a function f that has a global maximum at x = c and a relative
maximum at x = a, since f (c) is greater than f (x) for every value of x, while f (a) is only
greater than the value of f (x) for x near a. Since the function appears to decrease without
bound, f has no global minimum, though clearly f has a relative minimum at x = b.
Our emphasis in this section is on finding the global extreme values of a function (if
they exist). In so doing, we will either be interested in the behavior of the function over its
entire domain or on some restricted portion. The former situation is familiar and similar
to work that we did in the two preceding sections of the text. We explore this through a
particular example in the following preview activity.
Preview Activity 3.3. Let f (x) = 2 +
3
1 + (x + 1) 2 .
(a) Determine all of the critical numbers of f .
(b) Construct a first derivative sign chart for f and thus determine all intervals on
which f is increasing or decreasing.
(c) Does f have a global maximum? If so, why, and what is its value and where is the
maximum attained? If not, explain why.
(d) Determine lim
x→∞
f (x) and lim
x→−∞
f (x).
(e) Explain why f (x) > 2 for every value of x.
(f) Does f have a global minimum? If so, why, and what is its value and where is the
minimum attained? If not, explain why.
⊲⊳
Global Optimization
For the functions in Figure 3.17 and Preview Activity 3.3, we were interested in finding
the global minimum and global maximum on the entire domain, which turned out to be
(−∞, ∞) for each. At other times, our perspective on a function might be more focused due
to some restriction on its domain. For example, rather than considering f (x) = 2 +
3
1+(x+1) 2
for every value of x, perhaps instead we are only interested in those x for which 0 ≤ x ≤ 4,
and we would like to know which values of x in the interval [0, 4] produce the largest
possible and smallest possible values of f . We are accustomed to critical numbers playing
a key role in determining the location of extreme values of a function; now, by restricting
the domain to an interval, it makes sense that the endpoints of the interval will also be
important to consider, as we see in the following activity. When limiting ourselves to a
particular interval, we will often refer to the absolute maximum or minimum value, rather
than the global maximum or minimum.