Using the Candidates Test to Determine Absolute (Global) Extrema
Compare original-function values at all endpoints and interior critical numbers on a closed interval.
College Board Topic 5.5 · CED p. 98 (PDF p. 103) · CED effective Fall 2020; project course outline 2026–27.
On a closed finite interval, a continuous function’s absolute extrema can occur at endpoints or interior critical numbers. The candidates test lists those locations, evaluates f at every one, and compares the outputs. Differentiation supplies candidates; arithmetic on the original function selects the winners. This is why endpoints stay in the list even if their derivative is nonzero.
Do not discard nondifferentiable interior points. For a continuous absolute-value or piecewise function, a corner may be the smallest value. Do discard solutions of f′ = 0 outside the specified interval, and do not turn a domain hole into a candidate. Verify continuity on the interval before invoking the EVT-based search guarantee.
A calculator can solve f′ = 0 numerically and evaluate candidates for a complicated model. Record all roots in the interval and enough precision to compare values; graph windows and root searches can miss zeros, especially a tangent touch. Endpoint evaluation is still required. Equal largest or smallest outputs give multiple locations of the same absolute extremum.
Let's work through it
For f(x) = x³ − 3x on [−2,2], continuity holds and f′ = 3(x² − 1).
Why: A polynomial is continuous; the derivative finds interior candidates.
Candidates are endpoints −2,2 and critical numbers −1,1.
Why: Both candidate categories are required.
Values are f(−2) = −2, f(−1) = 2, f(1) = −2, f(2) = 2.
Why: Compare outputs of f, not slopes.
Maximum value 2 occurs at −1 and 2; minimum value −2 occurs at −2 and 1.
Why: Ties are legitimate; an absolute extremum need not have a unique location.
Try it from memory
Pause & retrieve
Try it before you peek.
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
Which function do you evaluate at candidates?
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
Which function do you evaluate at candidates?
Worked answer and reason
The original f.
The question asks for maximum/minimum output, not maximum/minimum slope.
Retrieval 2
What if the largest value occurs at two candidates?
Worked answer and reason
Report both locations and their common maximum value.
The EVT does not require uniqueness.
Catch the tempting mistake
- A positive second derivative proves an absolute minimum on any closed interval.
- It gives local information at an appropriate critical point; compare every candidate for closed-interval absolute extrema.
- Only derivative zeros are candidates.
- Include nondifferentiable interior domain points and endpoints.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
This calculator requires JavaScript. Use the physics formula sheets and show your working manually.
+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 5 problem 10 · No calculator · analytic
- Unit 5 problem 11 · No calculator · analytic
For the teacher
- Ask for the candidate list before any calculator evaluation.
- If a student has the correct winner but an incomplete list, have them repair the justification rather than treating a lucky comparison as complete.
Glossary and deeper reading · 5 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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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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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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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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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.
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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.