The extreme value theorem states that a continuous function over a closed, bounded interval has an absolute maximum and an absolute minimum. As shown in
[link] , one or both of these absolute extrema could occur at an endpoint. If an absolute extremum does not occur at an endpoint, however, it must occur at an interior point, in which case the absolute extremum is a local extremum. Therefore, by
[link] , the point
at which the local extremum occurs must be a critical point. We summarize this result in the following theorem.
Location of absolute extrema
Let
be a continuous function over a closed, bounded interval
The absolute maximum of
over
and the absolute minimum of
over
must occur at endpoints of
or at critical points of
in
With this idea in mind, let’s examine a procedure for locating absolute extrema.
Problem-solving strategy: locating absolute extrema over a closed interval
Consider a continuous function
defined over the closed interval
Evaluate
at the endpoints
and
Find all critical points of
that lie over the interval
and evaluate
at those critical points.
Compare all values found in (1) and (2). From
[link] , the absolute extrema must occur at endpoints or critical points. Therefore, the largest of these values is the absolute maximum of
The smallest of these values is the absolute minimum of
Now let’s look at how to use this strategy to find the absolute maximum and absolute minimum values for continuous functions.
Locating absolute extrema
For each of the following functions, find the absolute maximum and absolute minimum over the specified interval and state where those values occur.
over
over
Step 1. Evaluate
at the endpoints
and
Step 2. Since
is defined for all real numbers
Therefore, there are no critical points where the derivative is undefined. It remains to check where
Since
at
and
is in the interval
is a candidate for an absolute extremum of
over
We evaluate
and find
Step 3. We set up the following table to compare the values found in steps 1 and 2.
Conclusion
Absolute maximum
Absolute minimum
From the table, we find that the absolute maximum of
over the interval [1, 3] is
and it occurs at
The absolute minimum of
over the interval [1, 3] is
and it occurs at
as shown in the following graph.
This function has both an absolute maximum and an absolute minimum.
Step 1. Evaluate
at the endpoints
and
Step 2. The derivative of
is given by
for
The derivative is zero when
which implies
The derivative is undefined at
Therefore, the critical points of
are
The point
is an endpoint, so we already evaluated
in step 1. The point
is not in the interval of interest, so we need only evaluate
We find that
Step 3. We compare the values found in steps 1 and 2, in the following table.
Conclusion
Absolute maximum
Absolute minimum
We conclude that the absolute maximum of
over the interval [0, 2] is zero, and it occurs at
The absolute minimum is −2, and it occurs at
as shown in the following graph.
This function has an absolute maximum at an endpoint of the interval.
