A local (relative) maximum is a point where a function's value is greater than or equal to its value at all nearby points. On the AP Calculus exam, you find one at a critical point and justify it with the First Derivative Test (f' changes + to -) or the Second Derivative Test (f'(c) = 0 and f''(c) < 0).
A local maximum (the CED calls it a relative maximum) is a point where a function reaches a peak compared to everything nearby. Picture a hilltop on a hiking trail. It's the highest spot in your immediate area, even if a taller mountain exists somewhere else on the trail. That "somewhere else" part is exactly what separates a local max from an absolute max.
Local maxima can only happen at critical points, which are places where f'(x) = 0 or f'(x) doesn't exist. But here's the catch the AP exam loves to test. Not every critical point is a local max (think of y = x³ at x = 0, where the derivative is zero but nothing peaks). So finding f'(x) = 0 is step one, not the answer. You still have to prove the function actually switches from increasing to decreasing there, using either the sign change of f' or the concavity given by f''.
Local maxima live in Unit 5: Analytical Applications of Differentiation, and they tie together three CED learning objectives. AP Calc 5.3.A asks you to justify a function's behavior from its first derivative (f' goes from positive to negative at a local max). AP Calc 5.7.A brings in the second derivative as an alternate justification (f''(c) < 0 at a critical point means concave down, so it's a peak). And AP Calc 5.10.A puts it all to work in optimization, where "maximize the area" or "maximize the profit" really means "find and justify a local max." One especially exam-relevant piece of essential knowledge from 5.7: if a continuous function has only ONE critical point on an interval and it's a local extremum, it's automatically the absolute extremum there too. That single fact closes out a huge number of optimization FRQs.
Keep studying AP Calculus Unit 5
Visual cheatsheet
view galleryCritical Point (Unit 5)
Every local maximum sits at a critical point, but not every critical point is a local maximum. Critical points are the candidates; the derivative tests are the interview that decides which candidate actually gets the job.
Second Derivative Test (Unit 5)
This is the fast lane for classifying a critical point. If f'(c) = 0 and f''(c) < 0, the function is concave down at c, so c is a local max. Concave down means the curve is shaped like a frown, and the top of a frown is a peak.
Absolute Maximum and the Candidates Test (Unit 5)
A local max only wins its neighborhood. The absolute max wins the whole interval. The Candidates Test settles it by comparing every local extremum against the endpoint values, since on a closed interval the absolute max could be a peak or an endpoint.
Increasing and Decreasing Intervals (Unit 5)
A local max is literally the moment a function stops increasing and starts decreasing. That's why the First Derivative Test works. You're just checking that f' flips from positive to negative, which is Topic 5.3's sign analysis put to a specific use.
Multiple-choice questions hand you f' or f'' information (a formula, a graph, or a sign chart) and ask where local maxima occur, or they flip it and ask what f'(c) = 0 plus f''(c) < 0 tells you. Practice questions hit this from every angle, including what a zero derivative means at a point, how the First Derivative Test identifies local extrema, and what a negative second derivative indicates. On FRQs, the points come from your justification, not just the answer. Writing "x = 3 is a local max" earns nothing by itself. You need a sentence like "f has a local maximum at x = 3 because f' changes from positive to negative at x = 3." Watch for the classic trap question featuring a critical point that is NOT an extremum, like x³ at zero, designed to catch anyone who stops at f'(x) = 0.
A local maximum beats its neighbors; an absolute maximum beats the entire interval. A function can have several local maxima but at most one absolute maximum value. On a closed interval, the absolute max might not even be at a peak, since it can sit at an endpoint where the derivative tests say nothing. The one exception worth memorizing comes straight from the CED: if there's only one critical point on the interval and it's a local max, it's also the absolute max.
A local maximum is a point where the function's value is higher than at all nearby points, even if higher values exist elsewhere on the function.
Local maxima can only occur at critical points, where f'(x) = 0 or f'(x) is undefined, but a critical point alone does not guarantee a max.
The First Derivative Test confirms a local max when f' changes sign from positive to negative at the critical point.
The Second Derivative Test confirms a local max when f'(c) = 0 and f''(c) < 0, because the function is concave down there.
If a continuous function has exactly one critical point on an interval and it's a local max, it is also the absolute max on that interval.
On FRQs, you must state your reasoning explicitly, such as 'f' changes from positive to negative,' to earn the justification point.
A local maximum is a point where a function's value is greater than its value at all nearby points. It occurs at a critical point where the function switches from increasing to decreasing, and on the AP exam you prove it with the First or Second Derivative Test.
No. A zero derivative only gives you a critical point, which could be a local max, a local min, or neither. The classic counterexample is f(x) = x³ at x = 0, where f'(0) = 0 but the function never stops increasing.
A local max only needs to beat nearby points, while an absolute max is the single highest value on the whole interval. On a closed interval, the absolute max can also occur at an endpoint, which is why the Candidates Test compares critical points against endpoints.
Use one of two sentences. Either 'f has a local maximum at x = c because f' changes from positive to negative at x = c' (First Derivative Test), or 'f'(c) = 0 and f''(c) < 0, so f has a local maximum at x = c' (Second Derivative Test). Stating the conclusion without the sign reasoning loses the justification point.
Yes, the terms are interchangeable. The College Board CED uses 'relative (local) maximum,' so expect either word on exam questions and treat them identically.