---
title: "Relative Maximum — AP Calc Definition & Exam Guide"
description: "A relative maximum is where a function peaks compared to nearby points. Learn how the First and Second Derivative Tests prove it and how FRQs demand justification."
canonical: "https://fiveable.me/ap-calc/key-terms/relative-maximum"
type: "key-term"
subject: "AP Calculus AB/BC"
unit: "Unit 5"
---

# Relative Maximum — AP Calc Definition & Exam Guide

## Definition

A relative (local) maximum is a point where a function's value is greater than or equal to all nearby values. In AP Calculus, it occurs at a critical point where f' changes sign from positive to negative (FUN-4.A.2), or where f' = 0 and f'' < 0.

## What It Is

A relative maximum (also called a [local maximum](/ap-calc/key-terms/local-maximum "fv-autolink")) is a point where a [function](/ap-calc/unit-1/defining-continuity-at-point/study-guide/JbsR9iQfAzCznNOCG6JK "fv-autolink") is bigger than everything around it. Think of it as the top of one hill, not necessarily the tallest mountain on the whole graph. Formally, f has a relative maximum at x = c if f(c) ≥ f(x) for all x near c.

On the AP exam, the definition matters less than the *proof*. Relative maxima can only happen at critical points, places where f'(c) = 0 or f'(c) doesn't exist. But not every critical point is a max, so you confirm it one of two ways. The First Derivative Test ([Topic 5.4](/ap-calc/unit-5/using-first-derivative-test-to-determine-relative-local-extrema/study-guide/BjnQNCShz0uQiZGhSl2g "fv-autolink")) says f has a relative max at c if f' changes from positive to negative there (the function stops rising and starts falling). The Second Derivative Test (Topic 5.7) says if f'(c) = 0 and f''(c) < 0, the curve is concave down at c, so c is a relative max. Both tests live under the same essential knowledge: the first derivative of a function determines the location of relative extrema (FUN-4.A.2).

## Why It Matters

Relative maxima sit at the heart of [Unit 5](/ap-calc/unit-5 "fv-autolink") (Analytical Applications of Differentiation), specifically Topics 5.4, 5.7, and 5.9, all under the learning objective "Justify conclusions about the [behavior of a function](/ap-calc/key-terms/behavior-of-a-function "fv-autolink") based on the behavior of its derivatives" (5.4.A, 5.7.A, 5.9.A). That word "justify" is the whole game. The exam doesn't just ask *where* the max is; it asks you to write the sentence that proves it, usually "f' changes from positive to negative at x = c, so f has a relative maximum at x = c."

The term also crosses into Unit 6. When a function is defined as an accumulation function g(x) = ∫ from a to x of f(t) dt, then g' = f by the [Fundamental Theorem of Calculus](/ap-calc/unit-6/fundamental-theorem-calculus-definite-integrals/study-guide/fEGd7E9gbOH8EtCf "fv-autolink"), so finding relative maxima of g means reading sign changes off the graph of f (Topic 6.5, FUN-5.A.3). And one more high-value fact from FUN-4.A: if a continuous function has only one critical point on an interval and it's a relative max, it's automatically the absolute max on that interval.

## Connections

### First Derivative Test (Unit 5)

This is the most common way to prove a relative maximum. If f' switches from positive to negative at a [critical point](/ap-calc/key-terms/critical-point "fv-autolink"), the function climbed and then fell, so that point is a peak. The exact wording of that sign change is what earns the justification point on FRQs.

### Second Derivative Test (Unit 5)

The alternate proof. If f'(c) = 0 and f''(c) < 0, the graph is [concave down](/ap-calc/key-terms/concave-down "fv-autolink") at c (frowning shape), so c is a relative max. Watch the trap from practice questions: if f''(c) = 0, the test is inconclusive and you have to fall back on the First Derivative Test.

### [Accumulation Function (Unit 6)](/ap-calc/key-terms/accumulation-function)

When g(x) = ∫ from a to x of f(t) dt, the graph you're handed of f is actually the graph of g'. So g has a relative maximum exactly where f crosses from positive to negative. This is the bridge that makes Unit 5 logic reappear on Unit 6 problems like the 2023 FRQ Q4.

### [Critical Point (Unit 5)](/ap-calc/key-terms/critical-point)

Every relative maximum happens at a critical point, but the reverse isn't true. A critical point can be a max, a min, or neither (like f(x) = x³ at x = 0). Critical points are the candidates; the derivative tests are the interview.

## On the AP Exam

Multiple choice loves testing the logic of the two tests. Typical stems ask what you can conclude when f'' is zero at a critical point (answer: nothing yet, the Second Derivative Test is inconclusive) or what happens to an integrally-defined function h when its derivative changes from negative to positive (that's a relative *minimum*, the mirror-image trap). On the free response, relative maxima show up in graph-of-the-derivative problems like the 2023 FRQ Q4, where f' is given as line segments and a semicircle and you have to locate and justify extrema of f from sign changes. The 2021 FRQ Q5 pushed it further with implicit differentiation, asking whether a point on a curve was a relative max or min using the Second Derivative Test. In every case, the credited response requires a sign-based justification in words, not just an x-value. Saying "f' = 0 at x = c" alone never earns the point.

## Relative Maximum vs Absolute Maximum

A relative maximum beats its neighbors; an absolute maximum beats every point on the entire interval. A function can have several relative maxima but at most one absolute maximum value, and on a closed interval the absolute max might happen at an endpoint where no relative max exists at all. One important crossover from FUN-4.A: if a continuous function has exactly one critical point on an interval and it's a relative max, that point is also the absolute max. That fact is a fast justification shortcut on optimization FRQs.

## Key Takeaways

- A relative maximum is a point where the function value is greater than or equal to all nearby values, even if it's not the highest point on the whole graph.
- Relative maxima can only occur at critical points, where f' equals zero or is undefined, but not every critical point is a relative maximum.
- First Derivative Test: f has a relative maximum at x = c if f' changes from positive to negative at c, and you must state that sign change to earn justification points.
- Second Derivative Test: if f'(c) = 0 and f''(c) < 0, then c is a relative maximum; if f''(c) = 0, the test is inconclusive.
- For an accumulation function g(x) = ∫ from a to x of f(t) dt, relative maxima of g occur where the graph of f crosses from positive to negative, because f is g's derivative.
- If a continuous function has only one critical point on an interval and it's a relative maximum, it is automatically the absolute maximum on that interval.

## FAQs

### What is a relative maximum in AP Calculus?

It's a point where a function's value is greater than or equal to all nearby values, like the top of one hill on the graph. It occurs at a critical point where f' changes from positive to negative, or where f' = 0 and f'' < 0.

### Is every critical point a relative maximum?

No. A critical point is only a candidate. It could be a relative max, a relative min, or neither (f(x) = x³ has a critical point at x = 0 that's neither). You need the First or Second Derivative Test to classify it.

### What's the difference between a relative maximum and an absolute maximum?

A relative max only has to beat nearby points; an absolute max beats every point on the interval. On a closed interval, the absolute max can occur at an endpoint, which is never a relative max. The CED notes one shortcut: a single critical point that's a relative max on an interval is also the absolute max there.

### If f''(c) = 0 at a critical point, is it a relative maximum?

You can't tell. When the second derivative is zero at a critical point, the Second Derivative Test is inconclusive, and you have to check the sign change of f' instead. This is a classic multiple-choice trap.

### How do I find a relative maximum of an accumulation function like g(x) = ∫ f(t) dt?

By the Fundamental Theorem of Calculus, g' = f, so look at where the graph of f crosses from positive to negative. That x-value is a relative maximum of g. The 2023 FRQ Q4 tested exactly this skill with a graph of f' made of line segments and a semicircle.

## Related Study Guides

- [5.4 Using the First Derivative Test to Determine Relative (Local) Extrema](/ap-calc/unit-5/using-first-derivative-test-to-determine-relative-local-extrema/study-guide/BjnQNCShz0uQiZGhSl2g)
- [5.9 Connecting a Function, Its First Derivative, and its Second Derivative ](/ap-calc/unit-5/connecting-a-function-its-first-derivative-and-its-second-derivative/study-guide/NQXfonM48ssVKKRHQ7Te)
- [AP Calculus Free Response Question (FRQ) Overview](/ap-calc/exam-skills/frq-overview/study-guide/X0FYR1CY9RhaTPNYtL16)

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