Relative (local) extrema are points where a function's value is the highest or lowest compared to nearby points. On the AP Calc exam, every relative extremum occurs at a critical point (where f′ = 0 or f′ doesn't exist), but not every critical point is a relative extremum (FUN-1.C.3).
A relative extremum is a point where a function hits a peak or a valley compared to the points right around it. A relative maximum is a local peak, and a relative minimum is a local valley. The word "relative" (the CED also says "local") means you're only comparing the point to its neighbors, not to the whole graph. A function can have several relative maxima at different heights.
The rule that powers everything in Unit 5 is this one-way street: all relative extrema occur at critical points, which are places where f′(x) = 0 or f′ doesn't exist (FUN-1.C.2 and FUN-1.C.3). But the street only runs one direction. A critical point isn't automatically an extremum. Think of f(x) = x³ at x = 0. The derivative is zero there, but the graph just flattens out and keeps climbing, so there's no max or min. To confirm an actual extremum, you need the derivative to actually change sign, which is exactly what the First Derivative Test checks.
Relative extrema live in Topic 5.2 (Extreme Value Theorem, Global vs Local Extrema, and Critical Points) in Unit 5: Analytical Applications of Differentiation, supporting learning objective 5.2.A. This topic is the launching pad for the rest of Unit 5. Once you know extrema can only happen at critical points, you have a finite list of candidates to check, and the First Derivative Test, Candidates Test, and optimization problems all build on that idea. The exam loves this concept because it tests whether you understand what the derivative means, not just how to compute it. A correct justification ("f has a relative maximum at x = 2 because f′ changes from positive to negative there") is worth points; "f′(2) = 0" alone is not.
Keep studying AP Calculus Unit 6
Visual cheatsheet
view galleryCritical Point (Unit 5)
Critical points are the candidate list and relative extrema are the winners. You find where f′ = 0 or f′ doesn't exist first, then test each candidate to see if it's actually a max or min. Skipping the test is the classic way to lose justification points.
First Derivative Test (Unit 5)
This is the tool that confirms a relative extremum. If f′ changes from positive to negative at a critical point, you've got a relative max; negative to positive means a relative min. No sign change means no extremum, even though f′ = 0 there.
Absolute Extrema (Unit 5)
Absolute extrema are the single highest and lowest values on an entire interval, while relative extrema are just local peaks and valleys. On a closed interval, the absolute max or min can sit at an endpoint, and endpoints are never relative extrema in the AP framing. The Extreme Value Theorem (FUN-1.C.1) guarantees absolute extrema exist when f is continuous on [a, b].
First Derivative (Unit 2 → Unit 5)
Everything here is payoff from Unit 2. The derivative measures slope, so a peak is where slope flips from uphill to downhill. Relative extrema are where that geometric intuition becomes an exam-ready analytical procedure.
Relative extrema show up in multiple-choice questions that hand you f′ (as a formula, a graph, or a sign chart) and ask where f has a relative maximum or minimum. The classic trap answer is a critical point where f′ doesn't change sign. On FRQs, this concept appears in the standard "find and justify" task. You'll be asked to identify relative extrema and justify your answer, and the only justification that earns credit is a statement about f′ changing sign (or the Second Derivative Test). Practice questions on this term focus on two skills: distinguishing relative from absolute extrema, and identifying extrema by analyzing derivatives. So drill both the sign-chart analysis and the precise sentence you'd write to justify it.
Relative extrema are local. A relative max only beats its neighbors, like the tallest hill in your town. An absolute max beats every point on the interval, like Mount Everest. A function can have many relative extrema but at most one absolute maximum value on a closed interval, and that absolute max might occur at an endpoint, where relative extrema don't apply. On the exam, finding absolute extrema on [a, b] means comparing function values at critical points AND endpoints (the Candidates Test), while finding relative extrema means checking sign changes of f′ at critical points only.
Relative (local) extrema are points where a function is higher or lower than all nearby points, not necessarily the whole graph.
Every relative extremum occurs at a critical point, but not every critical point is a relative extremum (f(x) = x³ at x = 0 is the go-to counterexample).
A critical point is where f′(x) = 0 or f′(x) does not exist.
To confirm a relative extremum, show that f′ changes sign there: positive to negative means relative max, negative to positive means relative min.
Endpoints of a closed interval can be absolute extrema but are not considered relative extrema, so don't include them when listing relative max/min points.
On FRQs, write the full justification ("f′ changes from positive to negative at x = c"); stating f′(c) = 0 alone does not earn the point.
Relative extrema are points where a function reaches a local maximum or local minimum, meaning its value is the highest or lowest compared to nearby points. They appear in Topic 5.2 of Unit 5 and can only occur at critical points, where f′ = 0 or f′ doesn't exist.
No. All relative extrema occur at critical points, but the reverse fails. For f(x) = x³, f′(0) = 0 but x = 0 is not an extremum because f′ never changes sign. You must verify the sign change before claiming a max or min.
A relative extremum only needs to beat nearby points; an absolute extremum is the single highest or lowest value on the entire interval. On a closed interval [a, b], absolute extrema can occur at endpoints, but relative extrema can't, which is why the Candidates Test checks endpoints and critical points.
Find the critical points by solving f′(x) = 0 and noting where f′ doesn't exist, then check the sign of f′ on each side. If f′ goes positive to negative, you have a relative max; negative to positive means a relative min. This is the First Derivative Test.
Yes, the CED uses them interchangeably, writing "local (relative) extrema." College Board questions may use either word, so treat "local maximum" and "relative maximum" as identical.