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3.3. GLOBAL OPTIMIZATION
Activity 3.7.
Let g(x) =
1
3 x 3 − 2x + 2.
(a) Find all critical numbers of g that lie in the interval −2 ≤ x ≤ 3.
(b) Use a graphing utility to construct the graph of g on the interval −2 ≤ x ≤ 3.
(c) From the graph, determine the x-values at which the absolute minimum and
absolute maximum of g occur on the interval [−2, 3].
(d) How do your answers change if we instead consider the interval −2 ≤ x ≤ 2?
(e) What if we instead consider the interval −2 ≤ x ≤ 1?
⊳
In Activity 3.7, we saw how the absolute maximum and absolute minimum of a
function on a closed, bounded interval [a, b], depend not only on the critical numbers of
the function, but also on the selected values of a and b. These observations demonstrate
several important facts that hold much more generally. First, we state an important result
called the Extreme Value Theorem.
The Extreme Value Theorem: If f is a continuous function on a closed interval
[a, b], then f attains both an absolute minimum and absolute maximum on [a, b].
That is, for some value x m such that a ≤ x m ≤ b, it follows that f (x m ) ≤ f (x) for all
x in [a, b]. Similarly, there is a value x M in [a, b] such that f (x M ) ≥ f (x) for all x in
[a, b]. Letting m = f (x m ) and M = f (x M ), it follows that m ≤ f (x) ≤ M for all x in
[a, b].
The Extreme Value Theorem tells us that provided a function is continuous, on any
closed interval [a, b] the function has to achieve both an absolute minimum and an
absolute maximum. Note, however, that this result does not tell us where these extreme
values occur, but rather only that they must exist. As seen in the examples of Activity 3.7,
it is apparent that the only possible locations for relative extremes are either the endpoints
of the interval or at a critical number (the latter being where a relative minimum or
maximum could occur, which is a potential location for an absolute extreme). Thus,
we have the following approach to finding the absolute maximum and minimum of a
continuous function f on the interval [a, b]:
• find all critical numbers of f that lie in the interval;
• evaluate the function f at each critical number in the interval and at each endpoint
of the interval;
• from among the noted function values, the smallest is the absolute minimum of f
on the interval, while the largest is the absolute maximum.
3.3. GLOBAL OPTIMIZATION
Activity 3.7.
Let g(x) =
1
3 x 3 − 2x + 2.
(a) Find all critical numbers of g that lie in the interval −2 ≤ x ≤ 3.
(b) Use a graphing utility to construct the graph of g on the interval −2 ≤ x ≤ 3.
(c) From the graph, determine the x-values at which the absolute minimum and
absolute maximum of g occur on the interval [−2, 3].
(d) How do your answers change if we instead consider the interval −2 ≤ x ≤ 2?
(e) What if we instead consider the interval −2 ≤ x ≤ 1?
⊳
In Activity 3.7, we saw how the absolute maximum and absolute minimum of a
function on a closed, bounded interval [a, b], depend not only on the critical numbers of
the function, but also on the selected values of a and b. These observations demonstrate
several important facts that hold much more generally. First, we state an important result
called the Extreme Value Theorem.
The Extreme Value Theorem: If f is a continuous function on a closed interval
[a, b], then f attains both an absolute minimum and absolute maximum on [a, b].
That is, for some value x m such that a ≤ x m ≤ b, it follows that f (x m ) ≤ f (x) for all
x in [a, b]. Similarly, there is a value x M in [a, b] such that f (x M ) ≥ f (x) for all x in
[a, b]. Letting m = f (x m ) and M = f (x M ), it follows that m ≤ f (x) ≤ M for all x in
[a, b].
The Extreme Value Theorem tells us that provided a function is continuous, on any
closed interval [a, b] the function has to achieve both an absolute minimum and an
absolute maximum. Note, however, that this result does not tell us where these extreme
values occur, but rather only that they must exist. As seen in the examples of Activity 3.7,
it is apparent that the only possible locations for relative extremes are either the endpoints
of the interval or at a critical number (the latter being where a relative minimum or
maximum could occur, which is a potential location for an absolute extreme). Thus,
we have the following approach to finding the absolute maximum and minimum of a
continuous function f on the interval [a, b]:
• find all critical numbers of f that lie in the interval;
• evaluate the function f at each critical number in the interval and at each endpoint
of the interval;
• from among the noted function values, the smallest is the absolute minimum of f
on the interval, while the largest is the absolute maximum.