In AP Precalculus, a local (relative) maximum is a point where a polynomial function's output is greater than the outputs at nearby inputs, which happens where the function switches from increasing to decreasing, or possibly at an included endpoint of a restricted domain (EK 1.4.A.2).
A local maximum is a "peak" on a graph. At that point, the function's output value is bigger than the outputs at all the nearby input values. The official AP Precalc framing (EK 1.4.A.2) ties it to behavior, not just shape. A polynomial has a local maximum exactly where it switches from increasing to decreasing. Picture walking along the graph from left to right. You climb uphill, hit the top, and start heading downhill. That turning point is the local maximum.
Two details make this an AP-level idea instead of just a vocabulary word. First, if the domain is restricted, an included endpoint can also be a local extremum, even though the function never "turns around" there. Second, "local" only means the point beats its neighbors. A polynomial can have several local maxima, and the largest of all of them (if it exists) is the global maximum. Also useful for graph sketching: between every two distinct real zeros of a polynomial, there must be at least one local maximum or local minimum, because the function has to turn around to come back to the x-axis.
Local maximum lives in Topic 1.4 (Polynomial Functions and Rates of Change) in Unit 1, supporting learning objective 1.4.A, which asks you to identify key characteristics of polynomial functions related to rates of change. It's one of the core "behavior" words of the whole course. AP Precalc is built around describing how functions change, and local extrema are precisely the spots where increasing flips to decreasing. If you can find and justify local maxima from a graph, a table, or a description of where the function is increasing and decreasing, you've mastered the kind of reasoning Unit 1 keeps coming back to with rational functions, end behavior, and concavity.
Keep studying AP® Precalculus Unit 1
Local minimum (Unit 1)
The mirror image. A local minimum is where the function switches from decreasing to increasing, a valley instead of a peak. AP questions almost always make you sort out which turning points are which, so learn them as a pair.
Global maximum (Unit 1)
Every global maximum is a local maximum, but not the other way around. "Local" means the point beats its neighbors; "global" means it beats every output on the entire domain. On a restricted domain, the global max might sit at an endpoint rather than at a peak.
Real zero (Unit 1)
Zeros and extrema are linked by a guarantee from EK 1.4: between any two distinct real zeros, a polynomial must have at least one local extremum. The graph crosses the x-axis, has to turn around somewhere, and comes back. Exam questions use this to make you reason about extrema you can't see directly.
Point of inflection (Unit 1)
Don't mix these up. A local maximum is where increasing flips to decreasing (a peak). A point of inflection is where concavity flips, from curving up to curving down or vice versa. The function can still be increasing the whole time through an inflection point.
Local maxima show up mostly in multiple-choice, and the questions are rarely "point at the peak." Typical stems give you a polynomial on a restricted domain like [-3, 4] and ask you to identify all extrema, local and absolute, which forces you to check included endpoints, not just turning points. Another common setup gives you a degree-4 polynomial with four zeros and asks what must be true about its local extrema, testing whether you know a turning point has to sit between consecutive zeros. You'll also see questions that describe where a function is increasing and decreasing on different intervals and ask you to classify each transition point. The skill being graded is justification, stating that a local maximum exists because the function changes from increasing to decreasing there, or because an endpoint output beats nearby outputs.
A local maximum only needs to beat the outputs near it, so a graph can have several. A global (absolute) maximum is the single greatest output over the whole domain. Here's the trap on restricted-domain problems. The global max might be at an endpoint, while the tallest interior peak is only a local max. Always compare endpoint values against turning-point values before declaring anything "absolute."
A local maximum occurs where a polynomial function switches from increasing to decreasing, which is the official AP Precalc definition from EK 1.4.A.2.
On a restricted domain, an included endpoint can also count as a local extremum even though the function doesn't turn around there.
Every global maximum is a local maximum, but a local maximum is only guaranteed to beat nearby outputs, not all outputs.
Between any two distinct real zeros of a polynomial, there must be at least one local maximum or local minimum.
A polynomial of degree n has at most n - 1 local extrema, so a degree-4 polynomial with four real zeros has exactly three turning points.
Don't confuse a local maximum (where increasing flips to decreasing) with a point of inflection (where concavity flips).
It's a point where a polynomial's output is greater than the outputs at nearby inputs. It occurs where the function switches from increasing to decreasing, or possibly at an included endpoint of a restricted domain. This is Topic 1.4, EK 1.4.A.2.
No. A local maximum only has to be higher than the points right around it. The highest point on the entire domain is the global (absolute) maximum, and a graph can have multiple local maxima but at most one global maximum value.
Yes. When a polynomial is restricted to a domain like [-3, 4], an included endpoint can be a local extremum even though the function never changes direction there. AP questions on restricted domains expect you to check endpoints every time.
A local maximum is where the function changes from increasing to decreasing, a peak in the graph. A point of inflection is where the rate of change changes direction, meaning concavity flips. A function can pass through an inflection point while still increasing.
A degree-n polynomial has at most n - 1 local extrema total (maxima and minima combined). For example, a degree-4 polynomial with zeros at x = -3, 0, 2, and 5 must have exactly three local extrema, one between each pair of consecutive zeros.
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