Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
Distinguish attained absolute extrema, local extrema, and interior critical numbers from endpoints.
College Board Topic 5.2 · CED p. 95 (PDF p. 100) · CED effective Fall 2020; project course outline 2026–27.
A continuous function on a closed finite interval [a,b] reaches a largest and a smallest value. That is the Extreme Value Theorem. The updated CED explicitly uses the closed interval: values approached at a missing endpoint are not necessarily attained. The theorem guarantees extrema exist; it does not locate them or make them unique.
An absolute maximum is at least as high as every function value in the domain being considered. A local maximum is at least as high as nearby values; it need not win across the whole interval. Use point or x-value for the location and value for f(c). A critical number is a domain value where f′ is zero or does not exist. For interior extrema, either condition can matter: |x| has a local minimum at its corner, while x³ has derivative zero at zero without an extremum.
For a closed-interval absolute-extrema search, list interior critical numbers and endpoints separately. Endpoints are candidates because the domain stops there; they need not satisfy f′ = 0. A number outside the function’s domain is not a critical number. These distinctions keep a useful search method from becoming an incorrect definition.
For f(x) = 1/x on (0,1), continuity on its domain does not give an absolute maximum or minimum: values grow without bound near zero and approach 1 without reaching it near one. For f(x) = x on [0,1], both extrema occur at endpoints although f′ = 1 everywhere. The shape and the domain work together.
Let's work through it
Consider f(x) = |x − 2| on [0,5]. It is continuous on [0,5].
Why: EVT guarantees an absolute maximum and minimum.
The interior critical number is x = 2, where the derivative does not exist.
Why: A corner is a legitimate candidate if the function is defined there.
Compare f(0) = 2, f(2) = 0, f(5) = 3.
Why: Endpoints and interior critical numbers are separate candidate categories.
The minimum value is 0 at x = 2; the maximum value is 3 at x = 5.
Why: Absolute extrema compare actual output values throughout the interval.
Try it from memory
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Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
Is x = 0 a critical number for f(x) = 1/x?
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
Is x = 0 a critical number for f(x) = 1/x?
Worked answer and reason
No: f is not defined at zero.
A critical number must belong to the domain.
Retrieval 2
Must every critical point be an extremum?
Worked answer and reason
No; x³ has f′(0) = 0 but no local extremum there.
The derivative condition identifies candidates, not classifications.
Retrieval 3
Can EVT place the maximum at an endpoint?
Worked answer and reason
Yes.
Its conclusion concerns the entire closed interval.
Catch the tempting mistake
- A continuous function always has absolute extrema.
- EVT needs a closed bounded interval; continuity alone on an open or unbounded domain is insufficient.
- Every absolute-extrema candidate is an interior critical number.
- Endpoints also belong to the closed-interval candidate list.
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Put it to work
- Unit 5 problem 4 · No calculator · verbal
- Unit 5 problem 5 · No calculator · analytic
For the teacher
- Compare x on [0,1] and on (0,1) to expose why attained versus approached matters.
- Use “candidate” deliberately until a sign test or value comparison provides a conclusion.
Glossary and deeper reading · 6 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Continuity
A function is continuous at an input when its value exists, its finite limit exists, and those two values agree. At a domain endpoint, continuity uses the relevant one-sided approach. For example, x² is continuous at 3 because both its value and its nearby limit are 9. Continuity on an interval requires the appropriate condition at every input in that interval.
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Interval
An interval is a set of real numbers containing every number between any two of its members. Parentheses exclude an endpoint, while square brackets include it. For example, (1,4] includes 4 and every number between 1 and 4, but excludes 1. An infinite endpoint always uses a parenthesis because infinity is not a real endpoint.
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Absolute extremum
An absolute extremum is the greatest or least function value on the domain or interval under discussion. The comparison includes every allowed input, including included endpoints. For example, x² on [−1,2] has absolute maximum 4 at x=2 and absolute minimum 0 at x=0. An absolute extremum must be attained rather than merely approached.
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Function
A mathematical function assigns exactly one output to each input in its domain. Inputs can share an output, but one input cannot have two different outputs. For example, f(x)=x² assigns the output 9 to both 3 and −3. Function notation names the rule and its input; f(3) means the output when the input is 3.
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Critical point
A critical number is an input in the function’s domain where its derivative is zero or does not exist. The corresponding point on the graph is called a critical point. For example, x³ has a critical number at 0, but it has no local maximum or minimum there. Critical numbers are candidates for extrema and need further analysis; inputs outside the domain are excluded.
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Domain
The domain of a function is the set of inputs for which the function is defined. An expression and its context can both restrict the allowed inputs. For example, the real function f(x)=1/(x−2) has domain all real numbers except 2. For a model of elapsed time, a sensible domain may also require t≥0.