Determining Absolute Extrema on an Open or Unbounded Interval

Why Open and Unbounded Intervals Are Different

Finding absolute extrema on a closed interval is relatively systematic: find the critical points, evaluate the function at those points and at both endpoints, and compare the values.

That strategy depends on something important, though: the endpoints must actually belong to the interval. If an interval is open, an endpoint may be approached without ever being reached. If the interval is unbounded, there may not be an endpoint at all.

For example, consider the intervals , , and . The first has no included endpoints, the second has one included endpoint, and the third continues forever.

The key question is therefore not simply "where are the critical points?" The real question is:

A function can have a local maximum or minimum without having an absolute maximum or minimum on the interval.

The Extreme Value Theorem

The Extreme Value Theorem (EVT) gives a powerful guarantee about absolute extrema.

If a function is continuous on a closed interval , then the function must attain both an absolute maximum and an absolute minimum somewhere on that interval.

Notice the two conditions:

1. The function must be continuous.
2. The interval must be closed and bounded.

When the interval is open or unbounded, the EVT does not guarantee that absolute extrema exist.

The Basic Strategy

For an open, half-open, or unbounded interval, use the following approach:

1. Find all critical numbers in the domain.
2. Determine the behavior near excluded endpoints. Use one-sided limits when necessary.
3. Determine the behavior as or . This is especially important for unbounded intervals.
4. Evaluate the function at critical numbers that lie in the interval.
5. Compare all attainable values and limiting behavior.
6. State explicitly whether an absolute maximum or minimum exists.

A critical point is only a candidate. It becomes an absolute extremum only after you compare it with the rest of the interval.

Open Endpoints: The Endpoint Is Not Available

Suppose a function is defined on . The point is not part of the domain.

You may calculate the limit as , but that limiting value is not automatically an attainable function value.

A Limit Is Not Automatically an Extreme Value

This is one of the most important ideas in this topic.

If , that tells you what values the function approaches. It does not tell you that the function actually equals .

For an absolute maximum, the function must actually attain the largest value at some point in the domain.

Likewise, if the function approaches a value from above or below but never reaches it, that value is not an absolute minimum or maximum.

Half-Open Intervals

A half-open interval includes one endpoint but excludes the other. For example, includes but excludes .

The included endpoint must be treated like an ordinary candidate point. The excluded endpoint must instead be analyzed using a limit or other endpoint behavior.

Unbounded Intervals

An unbounded interval extends indefinitely in at least one direction. Examples include , , and .

For these intervals, you must investigate what happens as the independent variable becomes arbitrarily large or negative.

If the function grows without bound, then an absolute maximum cannot exist. If it decreases without bound, an absolute minimum cannot exist.

When There Is No Absolute Maximum or Minimum

Consider on the entire real line.

The derivative is , so the only critical number is . At that point, the function has a local maximum.

But examine the behavior as and :

The function becomes arbitrarily large in both directions. Therefore, the local maximum is actually not an absolute maximum — wait, the value at is 0, while the function becomes larger and larger as increases. This example demonstrates why the critical-point test alone is insufficient.

A Better Way to Think About Absolute Extrema

For an open or unbounded interval, think of the problem as a competition among three kinds of locations:

1. Critical numbers — places where the derivative is zero or undefined.
2. Included endpoints — endpoints that actually belong to the interval.
3. Boundary behavior — what happens near excluded endpoints or as .

The winner must be an attained value. A function approaching a value is not enough.

Worked Example: A Rational Function

Open Intervals Can Still Have Absolute Extrema

Do not make the opposite mistake and assume that an open interval means there are no absolute extrema.

For example, consider on . The interval does not contain its endpoints, but the function has a minimum at an interior point.

Common Mistakes

Practice Problems

The AP Calculus Takeaway

When the interval is closed and bounded, the Extreme Value Theorem guarantees that a continuous function has both an absolute maximum and an absolute minimum.

When the interval is open, half-open, or unbounded, that guarantee disappears. You must actively investigate the function's behavior.

The most important habit is simple: never stop after finding the critical points. Check the endpoints that actually belong to the interval, investigate excluded endpoints with limits, and examine what happens toward infinity on unbounded intervals.

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