What changes when an extremum candidate is a corner or an excluded endpoint?
You will be able to: Include derivative failures in the domain and reject comparisons with unavailable inputs.
What changes when an extremum candidate is a corner or an excluded endpoint?
The bottom of a V-shaped graph is a real lowest point even though its tangent slope is undefined. A hole at that same bottom would remove the value from the domain.
A useful starting point: How do you find the absolute extrema on a closed interval? →
Words and symbols before equations
- Nondifferentiable candidate
- A domain point where f′ fails to exist.
- Excluded endpoint
- A boundary not available as an input.
- One-sided approach
- Behavior as inputs approach a boundary from within the domain.
- Domain audit
- Checking which inputs actually belong before comparing values.
What this picture assumes
Original model; numerical readouts are rounded. Local extrema here mean interior extrema. Zero is in the domain of the first three examples; it is excluded for 1/x. Reciprocal curves are split at zero.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- x²: zero is stationary and a minimum; f′ changes negative to positive.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
For f=abs(x−1) on [−1,3], the corner x=1 is an interior candidate. Include it even though no equation f′=0 finds it. Compare the endpoint values 2 and the corner value 0.
For f=x on (0,1), neither endpoint belongs. The usual closed-interval candidate guarantee does not apply, and there is no largest or smallest attained value.
For f=1/x on [−1,1], the displayed interval contains a forbidden input. It is not a continuous function on that entire interval; values become unbounded near zero and no finite absolute extrema exist on its actual domain there.
Always list the function’s domain before solving derivative equations. Derivative singularities, function holes, corners and included endpoints are different cases.
A worked example, step by step
Find the absolute extrema of f=abs(x−2) on [0,5].
- The function is continuous throughout [0,5].
- Its derivative fails at the interior domain point x=2, which must be included.
- Candidate outputs are f(0)=2, f(2)=0 and f(5)=3.
- Minimum 0 occurs at 2; maximum 3 occurs at 5.
“Undefined derivative” does not mean “undefined function.” Conversely, a hole in f is never an attained-extremum candidate.
Would solving f′=0 find the minimum of abs(x−2)?
Compare with an explanation
No. Its derivative is never zero where it exists, so the corner must be included separately.
Predict. Change one thing. Explain.
Compare abs(x) with 1/x at zero. Explain why only the first has a candidate there, and connect that distinction to the graph’s markers.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
x²: zero is stationary and a minimum; f′ changes negative to positive.
Original model; numerical readouts are rounded. Local extrema here mean interior extrema. Zero is in the domain of the first three examples; it is excluded for 1/x. Reciprocal curves are split at zero.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the relevant derivative signs, theorem conditions, domain or geometric constraint. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionCompare the extrema of f=x² on [0,2] and on (0,2). Explain any difference without treating an excluded boundary as an input.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: On [0,2], continuity and the closed bounded domain guarantee extrema.
- 1 point: Minimum 0 is at 0; maximum 4 is at 2.
- 1 point: On (0,2), x² strictly increases and neither boundary is included.
- 1 point: No minimum or maximum is attained there, although outputs approach 0 and 4.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1Why check derivative failures?
They can be extrema at corners or cusps in the domain.
RECALL 2What does a hole remove?
The function value needed for an attained extremum at that input.
RECALL 3Is the closed-interval recipe automatic on an open domain?
No; boundary behavior and attainment need separate analysis.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
What changes when an extremum candidate is a corner or an excluded endpoint?
- Include interior corners when f is defined.
- Only compare values at inputs in the domain; inspect open boundaries separately.
Remember: “Undefined derivative” does not mean “undefined function.” Conversely, a hole in f is never an attained-extremum candidate.
Conditions: Original model; numerical readouts are rounded. Local extrema here mean interior extrema. Zero is in the domain of the first three examples; it is excluded for 1/x. Reciprocal curves are split at zero.
Refresh Kid · AP Calculus AB Unit 5 · Objectives FUN-4.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 5.5, FUN-4.A. CED effective Fall 2020, with Fall 2026 clarifications, checked September 17, 2026. Unit 5 has twelve official topics. Focused lesson titles and questions are original Refresh Kid teaching material.
Existence theorems require their stated hypotheses. Critical numbers must belong to the function’s domain; interior local extrema and included endpoints are handled explicitly. Extrema and inflection candidates need justification, not just a zero derivative. Sparse derivative samples do not establish interval-wide signs. Optimization includes the constraint, feasible domain and global comparison. Implicit-curve conclusions specify the branch and distinguish finite slopes from vertical tangents.
The Organic Chemistry Tutor Mean Value Theorem and Optimization Problems video titles, creator and relevant descriptions were checked; full videos were not reviewed. The free optimization video mentions additional paid material, which is not required here. Khan Academy’s unit destination was checked; its JavaScript lesson contents were not fully readable by the research tool. OpenStax Sections 4.3, 4.4, 4.5 and 4.7 were consulted for conceptual cross-checking. No creator scripts, questions, diagrams or artwork were copied. Refresh Kid is not affiliated with or endorsed by these providers.
GitHub’s 3D website collection and its camera-control example informed optional spatial inspection. The open-box model is original geometry using existing self-hosted Three.js with its MIT license. A 12 cm square sheet loses four equal corner squares. The remaining net folds into an open box; the geometry uses the same lengths as the labeled 2D net. Fold angle is a construction view, not an independent design variable; displayed volume refers to the fully upright box. Camera rotation changes no mathematical values. Complete 2D diagrams and explanations remain available without WebGL.
Independent teacher review and observation of students remain pending. Technical checks do not certify mathematical accuracy, accessibility or learning effectiveness. This is a review edition.
Optional official resource: Released AP Calculus AB questions and scoring guides. The archive spans multiple units; no entire exam question is assigned here.
Learn → Explore → Practice → Review is informed by the IES learning guide; this exact implementation has not been evaluated with learners.
Want to work through this with a tutor?
Bring your question about What changes when an extremum candidate is a corner or an excluded endpoint? Your explanation and answers remain free to access.