A global maximum (or absolute maximum) is the greatest output value a polynomial function attains over its entire domain. For polynomials, it exists only when the function has even degree and a negative leading coefficient, or when the domain is restricted to include an endpoint.
A global maximum is the single highest output value a function ever reaches. Compare that to a local maximum, which is just the highest point in its own neighborhood. A polynomial can have several local maxima, but the global maximum is the greatest of all of them. That's why it's also called the absolute maximum.
Here's the part the AP exam actually tests. Per essential knowledge 1.4.A.3 and 1.4.A.4, a polynomial of even degree with a negative leading coefficient must have a global maximum, because both ends of the graph plunge down to negative infinity, so somewhere in the middle there's a highest point. An odd-degree polynomial has neither a global maximum nor a global minimum, since one end always runs up to positive infinity and the other runs down to negative infinity. Restricted domains are the exception. If you chop the domain and include an endpoint, that endpoint can become the global maximum even if the unrestricted function would have none.
Global maximum lives in Topic 1.4 (Polynomial Functions and Rates of Change) in Unit 1, supporting learning objective 1.4.A, identifying key characteristics of polynomial functions related to rates of change. It's one of the cleanest examples of a big AP Precalc skill: predicting graph behavior straight from the analytical form without graphing anything. Read the degree, read the leading coefficient, and you instantly know whether a global max exists. That same degree-and-leading-coefficient reasoning powers end behavior questions later in Unit 1, so nailing it here pays off across the whole unit.
Keep studying AP® Precalculus Unit 1
Local maximum (Unit 1)
Every global maximum is also a local maximum, but not the other way around. Think of local maxima as the peaks of individual hills and the global maximum as the summit of the tallest one. The exam loves checking whether you know the difference.
Leading term (Unit 1)
The leading term is the global maximum's crystal ball. An even degree with a negative leading coefficient guarantees a global max; an odd degree guarantees there isn't one. You can answer existence questions without ever sketching the graph.
Local minimum (Unit 1)
Global extrema come as a mirrored pair of facts. Even degree with a positive leading coefficient forces a global minimum instead of a global maximum, since both ends shoot up and the graph bottoms out somewhere in between.
Real zero (Unit 1)
Zeros and extrema work together when you sketch. Between every two distinct real zeros, the polynomial has to turn around at least once, so the zeros tell you where local maxima and minima (and possibly the global max) must hide.
Global maximum shows up almost entirely as a logic question about degree and leading coefficient. Typical multiple-choice stems give you something like f(x) = x⁴ - 8x² + 12 and ask what you can conclude about global extrema (positive leading coefficient, even degree, so a global minimum exists but no global maximum). Others flip it: "f(x) = ax⁴ + ... has a global maximum, what must be true about a?" (a must be negative). Or they describe end behavior, like a degree-10 polynomial with lim(x→∞) P(x) = -∞, and ask how many global extrema it must have. Your job is to translate between analytical form, end behavior, and existence of extrema in both directions. No released FRQ has used the phrase verbatim, but the underlying reasoning supports any FRQ part asking you to justify features of a polynomial's graph from its equation.
A local maximum only beats the points right around it; a global maximum beats every point in the entire domain. A quartic with two humps has two local maxima, but only the taller hump is the global maximum (and only if the ends go down). On the exam, "global" or "absolute" means you must consider the whole domain, including end behavior, while "local" or "relative" means you only check the neighborhood.
A global maximum (also called the absolute maximum) is the greatest output value a polynomial reaches over its entire domain.
A polynomial has a global maximum only if it has even degree and a negative leading coefficient, because both ends of the graph fall to negative infinity.
An odd-degree polynomial has no global maximum and no global minimum, since its ends run in opposite directions.
Even degree with a positive leading coefficient gives a global minimum instead of a global maximum.
Every global maximum is also a local maximum, but a local maximum is only the global maximum if no other point on the graph is higher.
On a restricted domain, an included endpoint can serve as the global maximum even when the unrestricted polynomial would have none.
It's the greatest output value a polynomial function attains over its entire domain, also called the absolute maximum. For polynomials, it exists only when the degree is even and the leading coefficient is negative, or when an included endpoint on a restricted domain happens to be the highest point.
A local maximum is the highest point in its immediate neighborhood, while a global maximum is the highest point anywhere on the graph. A polynomial like x⁴ - 8x² + 12 can have a local maximum (at x = 0 here) without having any global maximum, since its ends rise to infinity.
No. Odd-degree polynomials never have one because one end always goes to positive infinity. Even-degree polynomials with a positive leading coefficient don't either; they have a global minimum instead. Only even degree plus a negative leading coefficient guarantees a global maximum.
Check just two things in the leading term. If the degree is even and the leading coefficient is negative, a global maximum must exist. This is exactly the reasoning AP Precalc multiple-choice questions test under learning objective 1.4.A.
Yes, the two names mean the same thing. The College Board's CED uses both terms, so expect either wording on the exam and treat them as interchangeable.
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