AP Calculus AB / Analytical Applications of Differentiation
Lesson
Concavity and the optimization write-up
Second derivatives describe bend. Worded max/min problems need a function, a domain, and a comparison.
Learning goals
- Use f″ to discuss concavity and inflection.
- Set up an optimization function and justify a maximum on a closed domain.
Explanation
If f″ is positive, the graph is concave up—slopes are increasing. If f″ is negative, concave down. An inflection point is a domain point where concavity changes. f″ = 0 is a clue, not a proof; the sign of f″ must actually switch.
Optimization starts with a primary equation (area, cost, distance) and a constraint that lets you write a single-variable function. State the realistic domain. Then use calculus candidates and, on a closed interval, the endpoints. A sentence with units finishes the job.
Key terms
- Inflection point. A point where a function changes concavity.
- Closed-interval method. Comparing f at critical points and endpoints to find absolute extrema on a closed interval.
Common mistakes
- Calling every zero of f″ an inflection point.
- Forgetting endpoints on a closed domain and missing the true maximum.
Practice
Original Marlow Works items. Check the answer explanation after you try.
What is lim (x→3) of (x² − 9) / (x − 3)?
Original Marlow Works item — not a College Board question.
Take your time—this is practice, not a test.
Answer explanation
Why must you still evaluate A at the endpoints when A′ has a single zero in (0, 20)?
Answer. On a closed interval the extreme value might occur at an end, so the critical point is only a candidate.
The first-derivative zero is not automatically global.
Related resources
External links with reuse status. Marlow Works is independent and does not copy restricted exam or textbook material.
Official / link only
AP Calculus AB course page
Official eight-unit framework and 2027 exam updates.
College Board · All rights reserved · accessed 2026-10-01
Open sourceReference only — do not copy
OpenStax College Algebra 2e
Algebra reference; current terms include noncommercial and AI-ingestion limits.
OpenStax · Restricted — see publisher page · accessed 2026-10-01
Open source