An absolute (global) maximum is the single largest y-value a function attains on its entire domain or a given interval. On the AP Calculus exam, you find it by comparing the function's values at critical points and endpoints, a process called the Candidates Test (LO 5.10.A).
An absolute maximum is the highest output a function ever reaches on the interval you're looking at. Not just the top of one hill, but the highest point on the whole map. If f(c) ≥ f(x) for every x in the domain, then f(c) is the absolute maximum. You'll also see it called the global maximum, and the two names are completely interchangeable.
Here's the part the exam actually tests. On a closed interval, the absolute maximum can only happen at two kinds of places: a critical point (where the first derivative is zero or undefined) or an endpoint of the interval. That short list of suspects is why the method is called the Candidates Test. You find the critical points, plug each candidate (critical points plus endpoints) into the original function, and the biggest output wins. This is the core skill behind LO 5.10.A, which asks you to calculate minimum and maximum values in applied contexts.
Absolute maximum lives primarily in Topic 5.10 (Introduction to Optimization Problems) in Unit 5, supporting LO 5.10.A: calculate minimum and maximum values in applied contexts or analysis of functions. The essential knowledge is blunt about it. The derivative exists to solve exactly this kind of problem, finding a max or min on a given interval. Optimization word problems (biggest area, cheapest fence, fastest route) are all absolute maximum or minimum problems in disguise.
It comes back in Topic 6.5 in Unit 6, where you analyze accumulation functions like g(x) = ∫ₐˣ f(t) dt (LO 6.5.A). Since g′(x) = f(x), finding where g has its absolute maximum means reading the graph of f, finding where it crosses zero, and running the Candidates Test using areas instead of algebra. That graph-based version is an FRQ classic.
Local Maximum / Relative Extrema (Unit 5)
A local maximum only beats its immediate neighbors; an absolute maximum beats every point on the interval. Every interior absolute max is also a local max, but most local maxes are not absolute. The exam loves checking that you know the difference.
Critical Point (Unit 5)
Critical points are where f′(x) = 0 or f′ is undefined, and they're your candidate list. The absolute maximum on a closed interval must occur at a critical point or an endpoint, so finding critical points is always step one.
Global Minimum (Unit 5)
The mirror image. Same Candidates Test, but now the smallest output wins. FRQs often ask for both extremes in one problem, so one pass through the candidates handles everything.
Accumulation Functions (Unit 6)
When g(x) = ∫ₐˣ f(t) dt, the FTC tells you g′ = f. So the absolute maximum of g sits where f changes from positive to negative, or at an endpoint, and you compare values by computing signed areas under the graph of f.
Multiple choice tends to test the concept directly. Questions ask what the candidates for an absolute maximum are, how a local max differs from an absolute max, or which theorem guarantees an extreme value exists on a closed interval (the Extreme Value Theorem, since the function is continuous there). Applied versions hand you a model like A(x) = 12x − x² for a channel's cross-section and ask you to justify that the critical point gives the absolute maximum.
FRQs raise the stakes with justification. The 2019 AB exam (Q3) and the 2026 AB/BC exams (Q4) both gave a graph of a derivative or of f and asked about extreme values of a function defined by an integral. To earn the points, you have to name your candidates, show the function's value at each one (often as a sum of areas), and explicitly compare them. Writing "x = 6 is a critical point" gets you nothing by itself. The justification "g(6) is greater than the values at every other critical point and endpoint, so it is the absolute maximum" is what scores.
A local (relative) maximum is the highest point in some small neighborhood, like the top of one hill. An absolute maximum is the highest point on the entire interval, the top of the tallest hill. A function can have several local maxes but at most one absolute maximum value, and on a closed interval the absolute max might sit at an endpoint, where it doesn't even count as a local max under the CED's definition. If a question says "justify that this is the absolute maximum," the First Derivative Test alone isn't enough; you must compare values at all candidates.
An absolute maximum is the largest value a function attains on its whole domain or a given interval, and it's the same thing as a global maximum.
On a closed interval, the absolute maximum can only occur at a critical point or an endpoint, so the Candidates Test means evaluating the function at each of those and picking the largest output.
The Extreme Value Theorem guarantees a continuous function on a closed interval actually has an absolute maximum and minimum, which is why the Candidates Test is allowed to work.
To justify an absolute maximum on an FRQ, you must compare the function's value at every candidate, not just show that the derivative changes sign at one point.
For an accumulation function g(x) = ∫ₐˣ f(t) dt, the absolute maximum of g occurs where f changes from positive to negative or at an endpoint, and you compare candidates using signed areas under f.
It's the single largest y-value a function reaches over its entire domain or a specified interval. You find it by evaluating the function at all critical points and endpoints (the Candidates Test) and choosing the biggest result, which is the skill behind LO 5.10.A.
A local maximum only needs to beat the points right around it, while an absolute maximum beats every point on the interval. A function can have multiple local maxes, but only one absolute maximum value, and it might land at an endpoint instead of a hilltop.
No. On a closed interval it can also occur at an endpoint, which is exactly why the Candidates Test includes endpoints. Skipping the endpoints is one of the most common ways to lose FRQ justification points.
On a closed interval, the candidates are the critical points (where f′(x) = 0 or f′ is undefined) and the two endpoints. Evaluate the original function at each candidate; the largest output is the absolute maximum.
List all candidates, compute the function's value at each, and state that your answer is the largest. On graph-based problems like 2019 AB Q3, those values come from signed areas, and graders want the explicit comparison written out, not just a derivative sign chart.