Extreme Value Theorem and Candidates Test

What the Extreme Value Theorem Is For

The Extreme Value Theorem (EVT) tells you when a continuous function is guaranteed to have both an absolute maximum and an absolute minimum on an interval. This is one of the most important ideas behind the Candidates Test.

The key conditions are simple: the function must be continuous, and the interval must be closed and bounded. If those conditions are satisfied, the function cannot avoid attaining its highest and lowest values somewhere on the interval.

Notice that the theorem guarantees that the extrema exist. It does not tell you where they occur. That's where the Candidates Test comes in.

The Extreme Value Theorem

If is continuous on the closed interval , then has both an absolute maximum and an absolute minimum on .

In practical AP Calculus problems, think:

• Continuous means there are no breaks, holes, jumps, or vertical asymptotes on the interval.
• Closed means both endpoints are included: .
• Bounded means the interval has finite endpoints.

Why the Endpoints Matter

One of the biggest AP Calculus mistakes is checking only where the derivative equals zero. That is not enough when you're looking for absolute extrema on a closed interval.

For a function on , the endpoints are always candidates. Even if the derivative does not equal zero there, the function can still achieve its absolute maximum or minimum at an endpoint.

The Candidates Test

The Candidates Test is the standard procedure for finding absolute maximum and minimum values of a continuous function on a closed interval.

1. Find the derivative .
2. Find all critical numbers in the interval. These occur where or where does not exist, provided the original function is defined there.
3. Evaluate the original function at every critical number in the interval.
4. Evaluate the original function at both endpoints.
5. Compare all candidate values. The largest output is the absolute maximum; the smallest output is the absolute minimum.

The crucial word is candidates. A critical number is not automatically a maximum or minimum. It is simply a point that might produce an absolute extremum.

Worked Example 1: Polynomial on a Closed Interval

Worked Example 2: A Critical Number That Is Not an Absolute Extremum

Critical Numbers: What Counts?

A critical number is a value in the domain of where either:

• , or
• does not exist.

For the Candidates Test, only critical numbers inside the interval are included as critical-number candidates. The endpoints are handled separately and must always be checked.

Worked Example 3: Derivative Undefined at a Critical Number

When the Extreme Value Theorem Does Not Apply

The EVT requires continuity on a closed interval. If either condition fails, the theorem does not guarantee absolute extrema.

Open interval

Consider on . The function is continuous, but the interval is open. The function approaches its largest possible value as , but that value is never actually attained.

Discontinuous function

A function with a hole or vertical asymptote on the interval is not continuous there. Therefore, the EVT cannot guarantee an absolute maximum or minimum.

Absolute vs. Local Extrema

Do not confuse an absolute maximum with a local maximum.

• An absolute maximum is the highest function value anywhere in the specified interval.
• An absolute minimum is the lowest function value anywhere in the specified interval.
• A local maximum is higher than nearby function values.
• A local minimum is lower than nearby function values.

A function can have several local extrema but only one absolute maximum value and one absolute minimum value on a given interval, although multiple points can sometimes share the same absolute value.

A Fast AP Exam Strategy

When an AP Calculus question asks for the absolute maximum or minimum of a function on a closed interval, use this checklist:

1. Write down the interval endpoints.
2. Differentiate the function.
3. Find every critical number in the interval.
4. Evaluate the original function at every critical number and both endpoints.
5. Compare the resulting function values.
6. State the answer clearly, including both the value and where it occurs when requested.

This method is reliable because every possible absolute extremum of a continuous function on a closed interval must occur at one of these candidates.

Common Mistakes

Practice Problems