---
title: "AP Calc AB/BC Unit 5 Review: Extrema & Curve Sketching"
description: "AP Calculus AB/BC Unit 5 covers Using the Mean Value Theorem and Determining Concavity. Study guides, practice questions, and key terms for every topic."
canonical: "https://fiveable.me/ap-calc/unit-5"
type: "unit"
subject: "AP Calculus AB/BC"
unit: "Unit 5 – Analytical Applications of Differentiation"
---

# AP Calc AB/BC Unit 5 Review: Extrema & Curve Sketching

## Overview

Unit 5 uses derivatives as reasoning tools rather than just computation tools. You apply the Mean Value Theorem and Extreme Value Theorem to justify conclusions, use sign charts for f' and f'' to classify critical points and describe concavity, and solve optimization problems by setting up an objective function and finding its extreme values. The unit closes with implicit relations, where the same derivative-based analysis extends to curves defined by equations in x and y.

## AP CED Alignment

This unit hub is organized around AP Course and Exam Description topics, skills, and exam task types when they are available in the source data.
- 5.1: Using the Mean Value Theorem
- 5.2: Extreme Value Theorem, Global vs. Local Extrema, and Critical Points
- 5.3: Determining Intervals on Which a Function Is Increasing or Decreasing
- 5.4: Using the First Derivative Test to Determine Relative (Local) Extrema
- 5.5: Using the Candidates Test to Determine Absolute (Global) Extrema
- 5.6: Determining Concavity of Functions over Their Domains
- 5.7: Using the Second Derivative Test to Determine Extrema
- 5.8: Sketching Graphs of Functions and Their Derivatives
- 5.9: Connecting a Function, Its First Derivative, and Its Second Derivative
- 5.10: Introduction to Optimization Problems
- 5.11: Solving Optimization Problems
- 5.12: Exploring Behaviors of Implicit Relations
- 5.1: Mean Value Theorem
- 5.2: Extreme Value Theorem and Critical Points
- 5.3-5.4: Increasing/Decreasing Intervals and the First Derivative Test
- 5.5: Candidates Test for Absolute Extrema
- 5.6-5.7: Concavity and the Second Derivative Test
- 5.8-5.9: Sketching and Connecting f, f', and f''
- 5.10-5.11: Setting Up and Solving Optimization Problems
- 5.12: Behaviors of Implicit Relations
- Practice 2 - Connecting Representations
- Practice 3 - Justification
- FRQs – No graphing calculator
- FRQs – Graphing calculator required

## Topics

- [5.1: Using the Mean Value Theorem](/ap-calc/unit-5/using-mean-value-theorem/study-guide/79sP2PXcyvRvBsjb3HRq): If f is continuous on [a,b] and differentiable on (a,b), the MVT guarantees a point c where f'(c) equals the average rate of change (f(b)-f(a))/(b-a). AP questions require you to verify conditions and state what the theorem guarantees.
- [5.2: Extreme Value Theorem, Global vs. Local Extrema, and Critical Points](/ap-calc/unit-5/extreme-value-theorem-global-vs-local-extrema-critical-points/study-guide/xcQI1ZzNbmWJ5uRNiFCo): The EVT guarantees absolute extrema for continuous functions on closed intervals. Critical points occur where f'(x) = 0 or f' is undefined. All local extrema are at critical points, but not all critical points are extrema.
- [5.3: Determining Intervals on Which a Function Is Increasing or Decreasing](/ap-calc/unit-5/determining-intervals-on-which-function-is-increasing-or-decreasing/study-guide/Y2fgjyl7H1dKPI2YsB4Y): Use a sign chart for f': positive f' means increasing, negative f' means decreasing. Partition the domain at critical points and domain breaks, then test the sign of f' in each interval.
- [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): At a critical point, if f' changes from positive to negative, f has a local max; from negative to positive, a local min; no sign change means no extremum. Always justify with the sign change of f'.
- [5.5: Using the Candidates Test to Determine Absolute (Global) Extrema](/ap-calc/unit-5/using-candidates-test-to-determine-absolute-global-extrema/study-guide/2ONEsyKKR6nyMs3UOpOZ): On a closed interval, collect all critical points and endpoints, evaluate f at each, and compare. The largest output is the absolute max; the smallest is the absolute min.
- [5.6: Determining Concavity of Functions over Their Domains](/ap-calc/unit-5/determining-concavity/study-guide/ORBIficQDT458eUIhJ0V): f''(x) > 0 means concave up; f''(x) < 0 means concave down. Inflection points occur where f'' changes sign. Build a sign chart for f'' to find concavity intervals.
- [5.7: Using the Second Derivative Test to Determine Extrema](/ap-calc/unit-5/using-second-derivative-test-determine-extrema/study-guide/g8rSveVmLeEuKLK4T7ep): At a critical point c where f'(c) = 0: f''(c) > 0 gives a local min, f''(c) < 0 gives a local max. If f''(c) = 0 or is undefined, the test is inconclusive; use the First Derivative Test instead.
- [5.8: Sketching Graphs of Functions and Their Derivatives](/ap-calc/unit-5/sketching-graphs-functions-their-derivatives/study-guide/aT1iYD0w3cZ4vq9YNoLG): Use the signs of f' and f'' to sketch f, or read a graph of f' to identify increasing/decreasing intervals, extrema, and inflection points of f. Corners and cusps on f appear as discontinuities on f'.
- [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): The graphs of f, f', and f'' are linked: zeros of f' with sign changes are extrema of f; zeros of f'' with sign changes are inflection points of f; local extrema of f' are inflection points of f.
- [5.10: Introduction to Optimization Problems](/ap-calc/unit-5/optimization-problems/study-guide/oepM07k8kwGY8zXZExoV): Optimization uses derivatives to find maximum or minimum values of a quantity. Write an objective function, use a constraint to reduce it to one variable, find critical points, and confirm the extremum type.
- [5.11: Solving Optimization Problems](/ap-calc/unit-5/solving-optimization-problems/study-guide/u2Y3MpOG6kkTtbLH38S7): Apply the full optimization process in applied contexts: define variables, write and reduce the objective function, find and confirm critical points, check endpoints on closed intervals, and interpret the answer with units.
- [5.12: Exploring Behaviors of Implicit Relations](/ap-calc/unit-5/exploring-behaviors-implicit-relations/study-guide/yRk3bNSDXIb7YidT78RL): Use implicit differentiation to find dy/dx for curves defined by F(x,y) = 0. Horizontal tangents occur where dy/dx = 0; vertical tangents where dy/dx is undefined. The second derivative, expressed in x, y, and dy/dx, determines concavity.

## Hardest Topics And Analytics

Snapshot: practice snapshot
This snapshot uses Fiveable practice activity to show where students tend to miss questions and which review moves are worth prioritizing first.
- **57% average MCQ accuracy** (Across 5.9k multiple-choice practice attempts for this unit.)
- **5.9k MCQ attempts** (Practice activity included in this snapshot.)
- **42% average FRQ score** (Across 8 scored free-response attempts for this unit.)
- **5.2: Extreme Value Theorem, Global vs. Local Extrema, and Critical Points**: 46% MCQ miss rate across 725 attempts. Review Extreme Value Theorem, Global vs. Local Extrema, and Critical Points with attention to how the concept appears in AP-style source and evidence questions.
- **5.9: Connecting a Function, Its First Derivative, and Its Second Derivative**: 45% MCQ miss rate across 463 attempts. Review Connecting a Function, Its First Derivative, and Its Second Derivative with attention to how the concept appears in AP-style source and evidence questions.
- **5.1: Using the Mean Value Theorem**: 38% MCQ miss rate across 815 attempts. Review Using the Mean Value Theorem with attention to how the concept appears in AP-style source and evidence questions.
- **5.4: Using the First Derivative Test to Determine Relative (Local) Extrema**: 38% MCQ miss rate across 533 attempts. Review Using the First Derivative Test to Determine Relative (Local) Extrema with attention to how the concept appears in AP-style source and evidence questions.

## Review Notes

### 5.1: Mean Value Theorem

The MVT states that if f is continuous on [a,b] and differentiable on (a,b), then there exists at least one c in (a,b) such that f'(c) = (f(b) - f(a)) / (b - a). Geometrically, the tangent line at c is parallel to the secant line through (a, f(a)) and (b, f(b)). On the AP exam, you must verify both conditions before invoking the theorem.

- **Continuity on [a,b]**: The function must have no breaks, jumps, or holes on the closed interval, including at the endpoints.
- **Differentiability on (a,b)**: The function must have a defined derivative at every point in the open interval; corners, cusps, and vertical tangents disqualify a point.
- **f'(c) = (f(b)-f(a))/(b-a)**: The guaranteed value c is where the instantaneous rate of change equals the average rate of change over [a,b].
- **Rolle's Theorem**: A special case of MVT where f(a) = f(b), guaranteeing at least one c where f'(c) = 0.

**Checkpoint:** Given f(x) = x^3 - 3x on [0,2], verify MVT conditions and find the value of c guaranteed by the theorem.

Theorem | Conditions | Conclusion
--- | --- | ---
Mean Value Theorem | Continuous on [a,b], differentiable on (a,b) | There exists c in (a,b) where f'(c) = (f(b)-f(a))/(b-a)
Rolle's Theorem | Continuous on [a,b], differentiable on (a,b), f(a)=f(b) | There exists c in (a,b) where f'(c) = 0
Extreme Value Theorem | Continuous on [a,b] | f has at least one absolute max and one absolute min on [a,b]

### 5.2: Extreme Value Theorem and Critical Points

The EVT guarantees that a function continuous on a closed interval [a,b] has both an absolute maximum and an absolute minimum on that interval. Critical points are where f'(x) = 0 or f'(x) is undefined. All local extrema occur at critical points, but not every critical point is a local extremum. A critical point with no sign change in f' is not an extremum.

- **Critical point**: An x-value where f'(x) = 0 or f'(x) does not exist; a necessary but not sufficient condition for a local extremum.
- **Local vs. absolute extrema**: A local extremum is the highest or lowest value in a neighborhood; an absolute extremum is the highest or lowest over the entire interval or domain.
- **EVT requirement**: Continuity on a closed interval is required; the theorem does not apply to open intervals or discontinuous functions.

**Checkpoint:** Explain why f(x) = 1/x on [-1,1] is not guaranteed an absolute maximum by the EVT.

Type of Extremum | Where it can occur | How to confirm
--- | --- | ---
Local maximum | Critical points only | f' changes from positive to negative
Local minimum | Critical points only | f' changes from negative to positive
Absolute maximum on [a,b] | Critical points or endpoints | Compare all candidate values
Absolute minimum on [a,b] | Critical points or endpoints | Compare all candidate values

### 5.3-5.4: Increasing/Decreasing Intervals and the First Derivative Test

To find where f is increasing or decreasing, locate all critical points and domain breaks, then test the sign of f' in each resulting interval. Where f'(x) > 0, f is increasing; where f'(x) < 0, f is decreasing. The First Derivative Test classifies each critical point: if f' changes from positive to negative, f has a local max; if f' changes from negative to positive, f has a local min; if the sign does not change, there is no extremum.

- **Sign chart for f'**: A number line partitioned at critical points and domain breaks, with the sign of f' recorded in each interval.
- **First Derivative Test**: Uses the sign change of f' at a critical point to classify it as a local max, local min, or neither.
- **No sign change**: If f' does not change sign at a critical point, that point is neither a local max nor a local min, even though f'(c) = 0.
- **Monotonicity**: A function is monotonically increasing on an interval if f'(x) > 0 throughout; monotonically decreasing if f'(x) < 0 throughout.

**Checkpoint:** For f(x) = x^4 - 4x^3, find all critical points, determine intervals of increase and decrease, and classify each critical point using the First Derivative Test.

Sign of f' before c | Sign of f' after c | Conclusion at c
--- | --- | ---
Positive | Negative | Local maximum
Negative | Positive | Local minimum
Positive | Positive | No extremum (increasing through c)
Negative | Negative | No extremum (decreasing through c)

### 5.5: Candidates Test for Absolute Extrema

On a closed interval [a,b], the absolute maximum and minimum can only occur at critical points or at the endpoints. The Candidates Test collects all critical points in (a,b) plus the two endpoints, evaluates f at each, and compares the output values. The largest output is the absolute maximum; the smallest is the absolute minimum. You compare f-values, not f'-values.

- **Candidates**: All critical points in the open interval (a,b) plus the endpoints a and b.
- **Evaluate f, not f'**: After finding candidates, plug each into the original function f to get output values for comparison.
- **EVT connection**: The Candidates Test works because the EVT guarantees the absolute extrema exist on a closed interval for a continuous function.

**Checkpoint:** Find the absolute maximum and minimum of f(x) = 2x^3 - 9x^2 + 12x on [0,3] using the Candidates Test.

Step | Action
--- | ---
1 | Find f'(x) and solve f'(x) = 0 or identify where f' is undefined
2 | List all critical points in (a,b) plus endpoints a and b
3 | Evaluate f at each candidate
4 | Compare values: largest is absolute max, smallest is absolute min

### 5.6-5.7: Concavity and the Second Derivative Test

The sign of f'' determines concavity: f''(x) > 0 means f is concave up on that interval (f' is increasing); f''(x) < 0 means f is concave down (f' is decreasing). An inflection point occurs where f'' changes sign. The Second Derivative Test classifies a critical point c: if f'(c) = 0 and f''(c) > 0, then f has a local min at c; if f''(c) < 0, then f has a local max at c. If f''(c) = 0 or is undefined, the test is inconclusive and you must use the First Derivative Test instead.

- **Concave up**: f''(x) > 0 on an interval; the graph curves upward and f' is increasing there.
- **Concave down**: f''(x) < 0 on an interval; the graph curves downward and f' is decreasing there.
- **Inflection point**: A point where f'' changes sign; the concavity of f switches at this x-value.
- **Second Derivative Test**: At a critical point c where f'(c) = 0: f''(c) > 0 gives a local min, f''(c) < 0 gives a local max, f''(c) = 0 is inconclusive.
- **Inconclusive case**: When f''(c) = 0 at a critical point, the Second Derivative Test gives no information; use the First Derivative Test.

**Checkpoint:** For f(x) = x^4, find all critical points and explain why the Second Derivative Test is inconclusive at x = 0. Then use the First Derivative Test to classify the point.

Test | What you need | Conclusion | When it fails
--- | --- | --- | ---
First Derivative Test | Sign of f' on both sides of c | Local max, min, or neither | Never fails, always gives an answer
Second Derivative Test | f'(c) = 0 and f''(c) defined and nonzero | Local max (f''<0) or local min (f''>0) | f''(c) = 0 or f''(c) undefined

### 5.8-5.9: Sketching and Connecting f, f', and f''

Topics 5.8 and 5.9 ask you to move between the graphs of f, f', and f'' in both directions. Where f has a local max or min, f' crosses zero with a sign change. Where f has an inflection point, f'' crosses zero with a sign change, and f' has a local extremum. Where f is increasing, f' is positive; where f is concave up, f'' is positive. A corner or cusp on f appears as a discontinuity or undefined point on f'.

- **Reading f from f'**: Positive f' means f is increasing; negative f' means f is decreasing; zeros of f' with sign changes are extrema of f.
- **Reading f from f''**: Positive f'' means f is concave up; negative f'' means concave down; zeros of f'' with sign changes are inflection points of f.
- **f' extrema and f inflection**: A local max or min of f' corresponds to an inflection point of f, because that is where f' stops increasing or decreasing.
- **Non-differentiable points**: A corner or cusp on f creates a jump or hole in the graph of f', and f'' is undefined there.

**Checkpoint:** Given a graph of f', identify all intervals where f is increasing, all local extrema of f, and all inflection points of f.

Feature of f | What it looks like on f' | What it looks like on f''
--- | --- | ---
Increasing | f' > 0 | No direct requirement
Local maximum | f' crosses zero from + to - | f'' < 0 (if Second Derivative Test applies)
Local minimum | f' crosses zero from - to + | f'' > 0 (if Second Derivative Test applies)
Inflection point | f' has a local extremum | f'' changes sign
Concave up | f' is increasing | f'' > 0

### 5.10-5.11: Setting Up and Solving Optimization Problems

Optimization problems ask you to find the maximum or minimum value of a quantity in a real context. The process: define variables and state what you are optimizing, write an objective function, use a constraint equation to reduce the objective function to one variable, determine the feasible domain, find critical points using f', and confirm whether each critical point is a max or min using the First or Second Derivative Test. Always check endpoints if the domain is a closed interval. Interpret the answer with correct units and in the context of the problem.

- **Objective function**: The function you want to maximize or minimize, written in terms of one variable after substituting the constraint.
- **Constraint equation**: A relationship between variables given by the problem context, used to eliminate one variable from the objective function.
- **Feasible domain**: The set of input values that make physical or contextual sense for the problem, which may restrict where you look for extrema.
- **Confirming max or min**: Use the First Derivative Test (sign change of f') or Second Derivative Test (sign of f'' at the critical point) to verify the nature of the critical point.
- **Interpret in context**: State the answer with units and explain what the maximum or minimum value means in the original problem situation.

**Checkpoint:** A farmer has 200 meters of fencing to enclose a rectangular field against a barn wall (one side needs no fencing). Write the objective function, find the dimensions that maximize area, and confirm it is a maximum.

Step | What to do
--- | ---
1. Define variables | Name the quantity to optimize and any other relevant quantities with units
2. Write objective function | Express the quantity to optimize as a function of your variables
3. Apply constraint | Use the given relationship to reduce the objective function to one variable
4. Find critical points | Differentiate and solve f'(x) = 0; note where f' is undefined
5. Confirm and interpret | Use First or Second Derivative Test; state the answer with units in context

### 5.12: Behaviors of Implicit Relations

For a curve defined by an equation F(x,y) = 0, you find dy/dx through implicit differentiation. Critical points occur where dy/dx = 0 (horizontal tangent) or where dy/dx is undefined (vertical tangent or cusp). To find the second derivative d2y/dx2 for an implicit relation, differentiate dy/dx implicitly again; the result is typically expressed in terms of x, y, and dy/dx. Use the sign of d2y/dx2 to determine concavity at a point, substituting the coordinates and the value of dy/dx at that point.

- **Horizontal tangent on implicit curve**: Occurs where dy/dx = 0; find by setting the numerator of dy/dx equal to zero (while the denominator is nonzero).
- **Vertical tangent on implicit curve**: Occurs where dy/dx is undefined; find by setting the denominator of dy/dx equal to zero (while the numerator is nonzero).
- **Second derivative of implicit relation**: d2y/dx2 is found by differentiating dy/dx implicitly; the result involves x, y, and dy/dx, so substitute all three at the point of interest.
- **Implicit function theorem**: Guarantees that an implicit relation can be treated locally as a function y = f(x) near a point where dy/dx is defined and the partial derivative with respect to y is nonzero.

**Checkpoint:** For x^2 + y^2 = 25, find all points with horizontal tangents, find all points with vertical tangents, and determine the concavity at (3,4) using the second derivative.

Feature | Condition on dy/dx | How to find it
--- | --- | ---
Horizontal tangent | dy/dx = 0 | Set numerator of dy/dx = 0, check denominator is nonzero
Vertical tangent | dy/dx undefined | Set denominator of dy/dx = 0, check numerator is nonzero
Concavity | Sign of d2y/dx2 | Differentiate dy/dx implicitly; substitute x, y, and dy/dx at the point

## Study Guides

- [5.1 Using the Mean Value Theorem](/ap-calc/unit-5/using-mean-value-theorem/study-guide/79sP2PXcyvRvBsjb3HRq)
- [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)
- [5.6 Determining Concavity](/ap-calc/unit-5/determining-concavity/study-guide/ORBIficQDT458eUIhJ0V)
- [5.3 Determining Intervals on Which a Function is Increasing or Decreasing](/ap-calc/unit-5/determining-intervals-on-which-function-is-increasing-or-decreasing/study-guide/Y2fgjyl7H1dKPI2YsB4Y)
- [5.7 Using the Second Derivative Test to Determine Extrema](/ap-calc/unit-5/using-second-derivative-test-determine-extrema/study-guide/g8rSveVmLeEuKLK4T7ep)
- [5.10 Introduction to Optimization Problems](/ap-calc/unit-5/optimization-problems/study-guide/oepM07k8kwGY8zXZExoV)
- [5.11 Solving Optimization Problems](/ap-calc/unit-5/solving-optimization-problems/study-guide/u2Y3MpOG6kkTtbLH38S7)
- [5.2 Extreme Value Theorem, Global vs Local Extrema, and Critical Points](/ap-calc/unit-5/extreme-value-theorem-global-vs-local-extrema-critical-points/study-guide/xcQI1ZzNbmWJ5uRNiFCo)
- [5.8 Sketching Graphs of Functions and Their Derivatives](/ap-calc/unit-5/sketching-graphs-functions-their-derivatives/study-guide/aT1iYD0w3cZ4vq9YNoLG)
- [5.5 Using the Candidates Test to Determine Absolute (Global) Extrema](/ap-calc/unit-5/using-candidates-test-to-determine-absolute-global-extrema/study-guide/2ONEsyKKR6nyMs3UOpOZ)
- [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)
- [5.12 Exploring Behaviors of Implicit Relations](/ap-calc/unit-5/exploring-behaviors-implicit-relations/study-guide/yRk3bNSDXIb7YidT78RL)

## Practice Preview

### Multiple-choice practice

- **AP-style practice question**: Practice 2 - Connecting Representations | The table below gives values of $$p(x)$$ and $$p'(x)$$ at selected $$x$$-values. At which $$x$$-value shown in the table is $$p$$ increasing most rapidly?

| $$x$$ | $$-2$$ | $$0$$ | $$2$$ | $$4$$ | $$6$$ |
|-------|--------|-------|-------|-------|-------|
| $$p(x)$$ | $$5$$ | $$8$$ | $$9$$ | $$7$$ | $$2$$ |
| $$p'(x)$$ | $$1.5$$ | $$2.8$$ | $$0.5$$ | $$-1.2$$ | $$-3.1$$ |
- **AP-style practice question**: Practice 3 - Justification | A particle's position along a line is given by $$s(t) = t^4 - 8t^3 + 18t^2$$ for $$t \geq 0$$. The velocity is $$v(t) = s'(t) = 4t^3 - 24t^2 + 36t = 4t(t-3)^2$$, with critical points at $$t = 0$$ and $$t = 3$$. Which principle determines whether these critical points represent local extrema of position?
- **AP-style practice question**: Practice 3 - Justification | A function $$f(x) = x^3 - 6x^2 + 9x + 2$$ is continuous on its domain. After finding that $$f'(x) = 3x^2 - 12x + 9 = 3(x-1)(x-3)$$, a student identifies critical points at $$x = 1$$ and $$x = 3$$. Which test should be applied to determine whether these critical points correspond to local maxima or minima?
- **AP-style practice question**: Practice 3 - Justification | A function $$f$$ is defined on the closed interval $$[1, 5]$$ by $$f(x) = \begin{cases} x^2 & \text{if } 1 \le x < 3 \\ 6 & \text{if } x = 3 \\ x + 1 & \text{if } 3 < x \le 5 \end{cases}$$. Which hypothesis required by the Extreme Value Theorem fails for $$f$$ on $$[1, 5]$$?
- **AP-style practice question**: Practice 3 - Justification | The function $$h(x) = \frac{1}{x - 2}$$ is being analyzed on the interval $$[0, 1]$$. Does the Extreme Value Theorem guarantee that $$h$$ has an absolute maximum and minimum on this interval? Justify your reasoning.
- **AP-style practice question**: Practice 2 - Connecting Representations | Consider the implicit relation $$x^3 + y^3 = 6xy$$ (a folium of Descartes). At a point where $$\frac{dy}{dx} = 0$$, which statement correctly describes the structural property of the curve?

### FRQ practice

- **Accumulation function, derivative, extrema, mean value theorem**: FRQs – No graphing calculator | Accumulation function, derivative, extrema, mean value theorem
- **Reservoir water level rates and accumulation**: FRQs – Graphing calculator required | Reservoir water level rates and accumulation

## Key Terms

- **Critical Points**: An x-value where f'(x) = 0 or f'(x) does not exist; a necessary but not sufficient condition for a local extremum of f.
- **Absolute Extrema**: The highest and lowest values a function reaches over a given interval; found using the Candidates Test on closed intervals.
- **Local Extrema**: The highest or lowest values of f within a small neighborhood of a point; all local extrema occur at critical points.
- **Candidates Test**: The procedure for finding absolute extrema on a closed interval: evaluate f at all critical points in (a,b) and at the endpoints a and b, then compare output values.
- **monotonicity**: The property of a function being entirely increasing (f' > 0) or entirely decreasing (f' < 0) on an interval, determined by the sign of the first derivative.
- **Concave Up**: A function is concave up on an interval where f''(x) > 0, meaning f' is increasing and the graph curves upward.
- **Concave Down**: A function is concave down on an interval where f''(x) < 0, meaning f' is decreasing and the graph curves downward.
- **Inflection Point**: A point on the graph of f where f'' changes sign, indicating a change in concavity from up to down or down to up.
- **Rolle's Theorem**: A special case of the MVT: if f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then there exists c in (a,b) where f'(c) = 0.
- **Average Rate of Change**: The slope of the secant line between two points on f, equal to (f(b)-f(a))/(b-a); the value that f'(c) equals at the MVT-guaranteed point c.
- **Instantaneous Rate of Change**: The value of f'(c) at a specific point c; geometrically, the slope of the tangent line to f at that point.
- **Reasoning with derivatives**: Using the sign and behavior of f' and f'' as evidence to draw justified conclusions about the behavior of f, such as increasing/decreasing intervals, extrema, and concavity.
- **implicit relation**: An equation relating x and y that is not solved explicitly for y; analyzed using implicit differentiation to find dy/dx and d2y/dx2.
- **Horizontal Tangent Line**: A tangent line with slope zero, occurring where f'(x) = 0 for explicit functions or where dy/dx = 0 for implicit relations.
- **Points of Inflection**: Locations on a graph where concavity changes; require f'' to change sign, not just equal zero.

## Common Mistakes

- **Forgetting to verify theorem conditions**: Applying the MVT or EVT without first confirming continuity (and differentiability for MVT) is an incomplete justification. Always check and state the conditions before citing the theorem's conclusion.
- **Treating every critical point as an extremum**: A critical point where f' does not change sign is not a local extremum. For example, f(x) = x^3 has f'(0) = 0, but x = 0 is not a local max or min because f' stays positive on both sides.
- **Using derivative values instead of function values in the Candidates Test**: When comparing candidates for absolute extrema, you must evaluate and compare f(x) values, not f'(x) values. The largest f-value is the absolute max; the smallest is the absolute min.
- **Declaring an inflection point wherever f'' = 0**: An inflection point requires f'' to change sign, not just equal zero. For f(x) = x^4, f''(0) = 0 but f'' does not change sign, so x = 0 is not an inflection point.
- **Forgetting to substitute dy/dx when evaluating d2y/dx2 for implicit relations**: The second derivative of an implicit relation is expressed in terms of x, y, and dy/dx. When evaluating concavity at a specific point, you must substitute the coordinates and the value of dy/dx at that point, not just x and y.

## Exam Connections

- **Justification with theorems and derivative evidence**: AP Calculus free-response questions in this unit frequently ask you to justify a conclusion rather than just compute an answer. A complete response names the theorem or test being used (MVT, EVT, First Derivative Test, Second Derivative Test), states the condition or sign observed, and writes the conclusion it supports. Partial credit is often lost when students state a correct answer without the required reasoning.
- **Graph-based reasoning across representations**: Multiple-choice and free-response questions often present information about f through a graph of f' or a table of values, then ask about features of f such as extrema, concavity, or inflection points. Practicing the translation between f, f', and f'' in both directions is essential for these tasks.
- **Optimization and implicit differentiation in applied contexts**: Free-response questions may present a real-world scenario requiring you to set up and solve an optimization problem, including writing the objective function, applying a constraint, and interpreting the result with units. Implicit differentiation questions may ask for the equation of a tangent line, the location of horizontal or vertical tangents, or the concavity of a curve at a specific point.

## Final Review Checklist

- **State MVT and EVT conditions correctly**: For both theorems, identify whether the function is continuous on [a,b] and, for MVT, differentiable on (a,b). State what each theorem guarantees before applying it.
- **Find all critical points accurately**: Set f'(x) = 0 and identify where f'(x) is undefined. Include domain restrictions. Remember that a critical point is not automatically an extremum.
- **Build and interpret sign charts for f' and f''**: Partition the domain at critical points and domain breaks. Test the sign of f' in each interval for increasing/decreasing behavior, and the sign of f'' for concavity. Use these charts to classify extrema and inflection points.
- **Apply the correct extrema test and know when each fails**: Use the Candidates Test for absolute extrema on closed intervals. Use the First Derivative Test for local extrema when the Second Derivative Test is inconclusive (f''(c) = 0 or undefined).
- **Set up optimization problems from scratch**: Define variables with units, write the objective function, substitute the constraint to get one variable, determine the feasible domain, and confirm whether the critical point gives a max or min.
- **Work with implicit relations using dy/dx and d2y/dx2**: Find dy/dx by implicit differentiation, locate horizontal and vertical tangents, and compute d2y/dx2 by differentiating dy/dx implicitly. Substitute x, y, and dy/dx at a specific point to evaluate concavity.
- **Justify conclusions in writing**: AP free-response scoring rewards explicit justification. Name the theorem or test you are using, state the sign or condition you observed, and write the conclusion it supports.

## Study Plan

- **Start with the theorems (5.1-5.2)**: Read the MVT and EVT topic guides and practice stating each theorem's conditions and conclusion in your own words. Work through two or three examples where you verify conditions, apply the theorem, and write a complete justification sentence.
- **Build sign chart fluency (5.3-5.5)**: Practice constructing sign charts for f' from scratch: find critical points, partition the domain, test signs, and write increasing/decreasing intervals. Then apply the First Derivative Test and Candidates Test on the same function to classify both local and absolute extrema.
- **Add concavity and the second derivative (5.6-5.7)**: Build sign charts for f'' to find concavity intervals and inflection points. Practice the Second Derivative Test and identify cases where it is inconclusive. Compare the First and Second Derivative Tests on the same critical point to see when each is more efficient.
- **Practice graph reading and sketching (5.8-5.9)**: Use the topic guides for 5.8 and 5.9 to practice translating between graphs of f, f', and f''. Given a graph of f', identify all features of f. Given a graph of f, sketch f'. Focus on the correspondence between zeros, sign changes, and extrema across all three graphs.
- **Work through optimization and implicit relations (5.10-5.12)**: Use the optimization topic guides to practice the full five-step process on classic problems (maximum area, minimum surface area, shortest distance). Then work through 5.12 examples finding horizontal and vertical tangents on implicit curves and computing d2y/dx2 at a specific point. Use available FRQ practice to rehearse written justification under exam conditions.

## More Ways To Review

- [Topic study guides](/ap-calc/unit-5#topics)
- [FRQ practice](/ap-calc/frq-practice)
- [Cram archive videos](/cram-archives?subject=ap-calculus&unit=unit-5)
- [Key terms](/ap-calc/key-terms)

## FAQs

### What topics are covered in AP Calc Unit 5?

AP Calc Unit 5 covers 12 topics focused on using derivatives to analyze function behavior. Topics include the Mean Value Theorem, the Extreme Value Theorem, critical points, intervals of increase and decrease, the First and Second Derivative Tests, concavity, sketching graphs of functions and their derivatives, and optimization problems. Implicit relations are covered in Topic 5.12. Here's the full topic list:
- 5.1 Using the Mean Value Theorem
- 5.2 Extreme Value Theorem, Global vs. Local Extrema, and Critical Points
- 5.3 Determining Intervals on Which a Function Is Increasing or Decreasing
- 5.4 Using the First Derivative Test to Determine Relative Extrema
- 5.5 Using the Candidates Test to Determine Absolute Extrema
- 5.6 Determining Concavity of Functions over Their Domains
- 5.7 Using the Second Derivative Test to Determine Extrema
- 5.8 Sketching Graphs of Functions and Their Derivatives
- 5.9 Connecting a Function, Its First Derivative, and Its Second Derivative
- 5.10 Introduction to Optimization Problems
- 5.11 Solving Optimization Problems
- 5.12 Exploring Behaviors of Implicit Relations See [AP Calc Unit 5](/ap-calc/unit-5) for notes and practice on each topic.

### How much of the AP Calc exam is Unit 5?

AP Calc Unit 5 makes up 15-18% of the AP exam, making it one of the more heavily tested units. It covers analytical applications of differentiation, including critical points, the First and Second Derivative Tests, concavity, optimization problems, and graph sketching. Expect several multiple-choice questions and at least one FRQ that draws on these skills.

### What's on the AP Calc Unit 5 progress check (MCQ and FRQ)?

The AP Calc Unit 5 progress check in AP Classroom includes both MCQ and FRQ parts drawn from the unit's 12 topics. The MCQ section tests skills like applying the Mean Value Theorem, identifying critical points, and using the First and Second Derivative Tests. The FRQ part asks you to justify conclusions about increasing and decreasing intervals, concavity, absolute and relative extrema, and optimization. The progress check also covers graph sketching and connecting a function to its first and second derivatives (Topics 5.8 and 5.9). Practicing these topics before the progress check is the best prep, and you can find matched practice at [AP Calc Unit 5](/ap-calc/unit-5).

### How do I practice AP Calc Unit 5 FRQs?

AP Calc Unit 5 FRQs most often ask you to analyze a function using its first and second derivatives, justify extrema with the First or Second Derivative Test, determine concavity, or solve an optimization problem. To practice, work through problems that require written justification, not just a numerical answer, because the AP exam awards points specifically for correct reasoning. Focus on Topics 5.3-5.7 for sign-chart justifications and Topics 5.10-5.11 for optimization setups. You can find Unit 5 FRQ-style practice at [AP Calc Unit 5](/ap-calc/unit-5).

### Where can I find AP Calc Unit 5 practice questions?

For AP Calc Unit 5 practice questions, including multiple-choice and practice test problems, head to [AP Calc Unit 5](/ap-calc/unit-5). You'll find topic-level MCQ practice covering the Mean Value Theorem, critical points, derivative tests, concavity, graph sketching, and optimization. Mixing MCQ practice with FRQ-style justification questions is the most effective way to prepare for the full range of question types this unit appears in on the exam.

### How should I study AP Calc Unit 5?

Start AP Calc Unit 5 by making sure you understand what a derivative tells you about a function, because every topic in this unit builds on that idea. Work through the topics in order: the Mean Value Theorem (5.1), then critical points and the Extreme Value Theorem (5.2), then increasing and decreasing intervals (5.3), then the First and Second Derivative Tests (5.4 and 5.7). Once those feel solid, move to concavity (5.6), graph sketching (5.8-5.9), and optimization (5.10-5.11). A few concrete steps that help:
- Draw sign charts for f' and f'' on every practice problem. Writing out the reasoning is exactly what earns points on FRQs.
- Practice justifying your answers in full sentences, not just circling a value.
- For optimization, always define your variables and write the constraint equation before taking a derivative.
- Review Topic 5.12 on implicit relations if you're in AP Calc BC or want extra challenge. All 12 topics with practice are at [AP Calc Unit 5](/ap-calc/unit-5).

## Structured Data

```json
{"@context":"https://schema.org","@type":"FAQPage","inLanguage":"en","mainEntity":[{"@type":"Question","@id":"https://fiveable.me/ap-calc/unit-5#what-topics-are-covered-in-ap-calc-unit-5","name":"What topics are covered in AP Calc Unit 5?","acceptedAnswer":{"@type":"Answer","text":"AP Calc Unit 5 covers 12 topics focused on using derivatives to analyze function behavior. Topics include the Mean Value Theorem, the Extreme Value Theorem, critical points, intervals of increase and decrease, the First and Second Derivative Tests, concavity, sketching graphs of functions and their derivatives, and optimization problems. Implicit relations are covered in Topic 5.12. Here's the full topic list:\n- 5.1 Using the Mean Value Theorem\n- 5.2 Extreme Value Theorem, Global vs. Local Extrema, and Critical Points\n- 5.3 Determining Intervals on Which a Function Is Increasing or Decreasing\n- 5.4 Using the First Derivative Test to Determine Relative Extrema\n- 5.5 Using the Candidates Test to Determine Absolute Extrema\n- 5.6 Determining Concavity of Functions over Their Domains\n- 5.7 Using the Second Derivative Test to Determine Extrema\n- 5.8 Sketching Graphs of Functions and Their Derivatives\n- 5.9 Connecting a Function, Its First Derivative, and Its Second Derivative\n- 5.10 Introduction to Optimization Problems\n- 5.11 Solving Optimization Problems\n- 5.12 Exploring Behaviors of Implicit Relations See <a href=\"/ap-calc/unit-5\">AP Calc Unit 5</a> for notes and practice on each topic."}},{"@type":"Question","@id":"https://fiveable.me/ap-calc/unit-5#how-much-of-the-ap-calc-exam-is-unit-5","name":"How much of the AP Calc exam is Unit 5?","acceptedAnswer":{"@type":"Answer","text":"AP Calc Unit 5 makes up 15-18% of the AP exam, making it one of the more heavily tested units. It covers analytical applications of differentiation, including critical points, the First and Second Derivative Tests, concavity, optimization problems, and graph sketching. Expect several multiple-choice questions and at least one FRQ that draws on these skills."}},{"@type":"Question","@id":"https://fiveable.me/ap-calc/unit-5#whats-on-the-ap-calc-unit-5-progress-check-mcq-and-frq","name":"What's on the AP Calc Unit 5 progress check (MCQ and FRQ)?","acceptedAnswer":{"@type":"Answer","text":"The AP Calc Unit 5 progress check in AP Classroom includes both MCQ and FRQ parts drawn from the unit's 12 topics. The MCQ section tests skills like applying the Mean Value Theorem, identifying critical points, and using the First and Second Derivative Tests. The FRQ part asks you to justify conclusions about increasing and decreasing intervals, concavity, absolute and relative extrema, and optimization. The progress check also covers graph sketching and connecting a function to its first and second derivatives (Topics 5.8 and 5.9). Practicing these topics before the progress check is the best prep, and you can find matched practice at <a href=\"/ap-calc/unit-5\">AP Calc Unit 5</a>."}},{"@type":"Question","@id":"https://fiveable.me/ap-calc/unit-5#how-do-i-practice-ap-calc-unit-5-frqs","name":"How do I practice AP Calc Unit 5 FRQs?","acceptedAnswer":{"@type":"Answer","text":"AP Calc Unit 5 FRQs most often ask you to analyze a function using its first and second derivatives, justify extrema with the First or Second Derivative Test, determine concavity, or solve an optimization problem. To practice, work through problems that require written justification, not just a numerical answer, because the AP exam awards points specifically for correct reasoning. Focus on Topics 5.3-5.7 for sign-chart justifications and Topics 5.10-5.11 for optimization setups. You can find Unit 5 FRQ-style practice at <a href=\"/ap-calc/unit-5\">AP Calc Unit 5</a>."}},{"@type":"Question","@id":"https://fiveable.me/ap-calc/unit-5#where-can-i-find-ap-calc-unit-5-practice-questions","name":"Where can I find AP Calc Unit 5 practice questions?","acceptedAnswer":{"@type":"Answer","text":"For AP Calc Unit 5 practice questions, including multiple-choice and practice test problems, head to <a href=\"/ap-calc/unit-5\">AP Calc Unit 5</a>. You'll find topic-level MCQ practice covering the Mean Value Theorem, critical points, derivative tests, concavity, graph sketching, and optimization. Mixing MCQ practice with FRQ-style justification questions is the most effective way to prepare for the full range of question types this unit appears in on the exam."}},{"@type":"Question","@id":"https://fiveable.me/ap-calc/unit-5#how-should-i-study-ap-calc-unit-5","name":"How should I study AP Calc Unit 5?","acceptedAnswer":{"@type":"Answer","text":"Start AP Calc Unit 5 by making sure you understand what a derivative tells you about a function, because every topic in this unit builds on that idea. Work through the topics in order: the Mean Value Theorem (5.1), then critical points and the Extreme Value Theorem (5.2), then increasing and decreasing intervals (5.3), then the First and Second Derivative Tests (5.4 and 5.7). Once those feel solid, move to concavity (5.6), graph sketching (5.8-5.9), and optimization (5.10-5.11). A few concrete steps that help:\n- Draw sign charts for f' and f'' on every practice problem. Writing out the reasoning is exactly what earns points on FRQs.\n- Practice justifying your answers in full sentences, not just circling a value.\n- For optimization, always define your variables and write the constraint equation before taking a derivative.\n- Review Topic 5.12 on implicit relations if you're in AP Calc BC or want extra challenge. All 12 topics with practice are at <a href=\"/ap-calc/unit-5\">AP Calc Unit 5</a>."}}]}
```