In AP Calculus, a local (relative) extremum is a point where a function's value is the highest (local maximum) or lowest (local minimum) compared to all nearby points; every local extremum occurs at a critical point, where f' equals zero or fails to exist (FUN-1.C.3).
A local extremum is a spot where the function is bigger or smaller than everything immediately around it. Think of a hiking trail. A local maximum is the top of any hill, even a small one, and a local minimum is the bottom of any dip. It doesn't have to be the tallest peak on the whole trail. It just has to beat its neighbors.
The CED ties this directly to critical points. A critical point is any place where the first derivative equals zero or doesn't exist (FUN-1.C.2). The key fact, and the one the exam loves to test, is the one-way relationship in FUN-1.C.3. All local extrema occur at critical points, but not all critical points are local extrema. The classic counterexample is f(x) = x³ at x = 0. The derivative is zero there, so it's a critical point, but the function never turns around, so there's no extremum. Critical points are candidates, not guarantees.
Local extrema live in Unit 5: Analytical Applications of Differentiation, specifically Topic 5.2, and support learning objective 5.2.A, justifying conclusions about functions using the Extreme Value Theorem. This concept is the gateway to almost everything else in Unit 5. The First Derivative Test, the Candidates Test for absolute extrema, and optimization problems all start with the same move. You find critical points, then decide which ones are actually local extrema. If you blur the line between "critical point" and "local extremum," you'll lose justification points on FRQs, because the College Board specifically rewards (and penalizes) the logic in that distinction.
Keep studying AP Calculus Unit 5
Visual cheatsheet
view galleryCritical Point (Unit 5)
This is the closest relationship on the page. Critical points are the full candidate pool, and local extrema are the candidates that actually win. f' = 0 or f' undefined gets a point into the pool, but you need a sign change in f' to confirm an extremum.
Extreme Value Theorem (Unit 5)
The EVT guarantees a continuous function on a closed interval [a, b] has an absolute max and min (FUN-1.C.1). Those absolute extrema occur either at local extrema inside the interval or at the endpoints, which is exactly why the Candidates Test checks both.
First Derivative (Units 2-5)
The derivative you learned to compute in Units 2 and 3 becomes a detection tool here. Where f' switches from positive to negative, you have a local max. Negative to positive means a local min. No sign change means no extremum, just a flat spot or a corner.
Relative Maximum and Relative Minimum (Unit 5)
These are the two flavors of local extremum, and the CED uses "local" and "relative" interchangeably. If a question says "relative max," it's asking about exactly the same thing as a local max, so don't let the vocabulary swap throw you.
Local extrema show up constantly in multiple choice, usually in two stem patterns. One gives you critical points and asks which are actually local extrema, like a question on h(x) = x⁴ - 4x³ with critical points at x = 0 and x = 3 (only x = 3 is an extremum, since f' doesn't change sign at x = 0). The other tests the logic itself, such as "a function has exactly three critical points and exactly two are local extrema; what must be true?" You need to know the candidate-versus-winner relationship cold. On FRQs, this concept appears in justification language. Writing "f has a local minimum at x = 3 because f' changes from negative to positive at x = 3" earns the point. Writing "because f'(3) = 0" does not, since that alone only proves a critical point. Expect it paired with the EVT too, where you reason from a single local extremum on an open interval to conclusions about absolute extrema on the closed interval.
A local extremum only has to beat its immediate neighbors, while a global extremum has to beat every point in the entire interval or domain. A global extremum can also be a local one (the tallest hill is still a hill), but plenty of local extrema are just small bumps that lose to an endpoint or a bigger peak elsewhere. On a closed interval, absolute extrema can occur at endpoints, which are usually not counted as local extrema, so finding local extrema alone is not enough to answer an absolute extrema question.
A local extremum is a point where the function is higher (local max) or lower (local min) than all nearby points, and "local" and "relative" mean the same thing on the AP exam.
All local extrema occur at critical points, but not all critical points are local extrema, which is exactly what FUN-1.C.3 says and what MCQs repeatedly test.
f(x) = x³ at x = 0 is the go-to counterexample of a critical point that is not a local extremum, because f' is zero there but never changes sign.
To justify a local extremum on an FRQ, you must show that f' changes sign at the critical point, not just that f' equals zero.
A global extremum can also be a local extremum, but on a closed interval the absolute max or min might sit at an endpoint instead, so always check endpoints separately.
A local extremum is a point where a function's value is the highest or lowest compared to all nearby points, making it a local maximum or local minimum. The CED (FUN-1.C.3) guarantees every local extremum happens at a critical point, where f' is zero or undefined.
No. Critical points are only candidates. f(x) = x³ has a critical point at x = 0 where f'(0) = 0, but the function keeps increasing through it, so there's no extremum. You need f' to change sign for a critical point to be a local extremum.
A local extremum only beats nearby points, while a global (absolute) extremum beats every point on the interval. On a closed interval [a, b], the absolute max or min can occur at an endpoint, which is why the Candidates Test checks both critical points and endpoints.
Yes. If the highest point on the whole interval happens to sit at an interior critical point, it's both a local and a global maximum at once. The tallest peak on the trail is still a peak.
Yes, they're identical. The College Board CED writes it as "local (relative) extrema," so a question about a relative maximum is asking about a local maximum. Don't second-guess the wording on the exam.