In AP Calculus, local (relative) extrema are points where a function reaches a high or low compared to nearby points. They can only occur at critical points (where f' is zero or undefined), and you confirm them by showing f' changes sign there.
A local extremum is a point where the function is bigger (local maximum) or smaller (local minimum) than everything immediately around it. Think of a hiking trail. A local max is the top of any hill on the trail, even a small one. The absolute max is the single tallest peak on the whole hike. Every peak counts as local, but only one is the highest overall.
The CED pins down exactly where these can happen. All local extrema occur at critical points, which are places where the first derivative equals zero or fails to exist (FUN-1.C.2 and FUN-1.C.3). But the arrow only points one way. Not every critical point is a local extremum. A classic example is f(x) = x³ at x = 0, where f'(0) = 0 but the graph just flattens and keeps climbing. To actually confirm a local extremum, you show the derivative changes sign there. If f' goes from positive to negative, you have a local max. If f' goes from negative to positive, you have a local min.
Local extrema live in Unit 5 (Analytical Applications of Differentiation), specifically Topics 5.2, 5.3, and 5.10. They sit at the center of three learning objectives. AP Calc 5.2.A has you apply the Extreme Value Theorem and the definition of critical points. AP Calc 5.3.A asks you to justify conclusions about a function from the behavior of its derivative, which is exactly what identifying a local max or min requires. AP Calc 5.10.A uses the same machinery to calculate minimum and maximum values in applied optimization problems. If Unit 5 had a main character, it would be the local extremum, because almost every justification question in the unit is really asking 'where does f' change sign, and what does that tell you about f?'
Keep studying AP Calculus Unit 5
Visual cheatsheet
view galleryCritical Points (Unit 5)
Critical points are the candidate list and local extrema are the winners. Per FUN-1.C.3, every local extremum sits at a critical point, but a critical point can also be a flat spot that's neither max nor min. Finding f' = 0 is step one, never the whole answer.
First Derivative Test (Unit 5)
This is the tool that actually classifies a local extremum. You check the sign of f' on each side of a critical point. Positive to negative means local max, negative to positive means local min. On FRQs, the sign change IS the justification graders look for.
Absolute Extrema and the Candidates Test (Unit 5)
On a closed interval, the Extreme Value Theorem (FUN-1.C.1) guarantees an absolute max and min exist. To find them, you compare f at every critical point and at both endpoints. Local extrema feed into this list, but an endpoint can beat them all.
Optimization Problems (Unit 5)
Topic 5.10 is local extrema wearing a word-problem costume. Maximizing area or minimizing cost means building a function, finding its critical points, and verifying you've got the right kind of extremum. Same skill, applied context.
Multiple choice loves to hand you a graph or table of f' and ask where f has a local maximum or minimum. The trap answer is always a point where f' = 0 but doesn't change sign. On FRQs, local extrema show up as justification questions, and the rubric wants the reason, not just the location. Writing 'f has a local minimum at x = 2 because f' changes from negative to positive at x = 2' earns the point. Writing 'because f'(2) = 0' does not. Practice questions also push the local-versus-global distinction, like whether a local extremum can also be a global one (yes, it can) and when a problem demands global extrema instead, which happens any time you're optimizing over a closed interval and need the Candidates Test.
Local extrema are highs and lows relative to nearby points, so a function can have several. Absolute extrema are the single highest and lowest values on the entire interval, and on a closed interval they can occur at endpoints where f' never changes sign at all. A local max can also be the absolute max, but you only know after comparing it against every other candidate, endpoints included.
Local extrema can only occur at critical points, which are where f' equals zero or doesn't exist.
Not every critical point is a local extremum, so f'(c) = 0 alone never proves a max or min (think f(x) = x³ at x = 0).
A local maximum happens where f' changes from positive to negative, and a local minimum happens where f' changes from negative to positive.
On an FRQ, the justification that earns the point is the sign change of f', not just the fact that the derivative equals zero.
A local extremum can also be the absolute extremum, but on a closed interval you must compare it against the endpoints using the Candidates Test.
The Extreme Value Theorem guarantees absolute extrema on a closed interval when f is continuous, but it says nothing about where local extrema are.
Local extrema are points where a function hits a high (local max) or low (local min) compared to the points immediately around it. They occur only at critical points, where the first derivative is zero or undefined, and you confirm them by showing f' changes sign.
No. The CED states all local extrema occur at critical points, but not all critical points are local extrema. For f(x) = x³, the derivative is zero at x = 0, yet the graph just flattens momentarily and keeps increasing, so there's no max or min there.
Local (relative) extrema are highs and lows compared to nearby points, and a function can have many of them. Absolute (global) extrema are the single highest and lowest values over the whole interval, and on a closed interval they might occur at an endpoint instead of a critical point.
Yes. If the highest local maximum is also bigger than the function's values at the endpoints (and everywhere else on the interval), it's the absolute maximum too. The Candidates Test is how you check.
State that f' changes from positive to negative at that x-value. Saying only that f'(c) = 0 won't earn the justification point, because a zero derivative without a sign change doesn't guarantee an extremum.