AP Calculus ABAP Exams

AP Calculus AB Unit 5 Review: The Most-Tested Extrema and Concavity Questions

Sigma Prep

Math Instructor · 10+ Years Experience

October 24, 2026·14 min read

Key takeaways

Unit weight: the biggest multiple choice unit, 65 of 315 official questions (20.6%), and 15.0% of the free-response points.
Top types: concavity and extrema from the graph of $f'$ , the candidates test, and increasing and decreasing intervals.
Practice: reading the graph of $f'$ . It was on the free response of 11 of the 15 exams since 2012.

Unit 5 is the biggest unit of the AP Calculus AB multiple choice. College Board’s Course and Exam Description (the CED) gives Unit 5 15–20% of the multiple choice. On the 7 official practice exams from 2012–2018 it was 65 of 315 questions (20.6%), between 8 and 12 per exam, just above that range and more than any other unit. On the 42-question section that starts in May 2027, that rate is about 9 questions.

It counts on the free response too. Unit 5 earned 15.0% of the points on the 2012–2026 exams, and 34 of the 86 free-response questions since 2012 had a Unit 5 part, on every one of the 15 exams. The common ones:

Reading the graph of $f'$ (11 questions, on 11 of the 15 exams): where $f$ increases, its relative and absolute extrema, its concavity and points of inflection.
Implicit curves (10 questions, 2012–2025): the second derivative $\frac{d^2y}{dx^2}$ at a point, and where the tangent line is horizontal or vertical.
The Mean Value Theorem (8 questions), 7 of them a justification from given values: must $f'(c)$ equal the average rate of change somewhere?
Analysis from a formula (9 questions): critical points, intervals of increase, the candidates test and the second derivative test.

The 12 most-tested Unit 5 question types

Ranked by share of the whole exam. The twelve add up to about 13.2% of the exam, out of 18.0% for all of Unit 5. Nine smaller types follow them.

1. Concavity and inflection from the graph of $f'$

The graph of $f$ is concave up where $f'$ is increasing and concave down where $f'$ is decreasing. A point of inflection is where $f'$ turns around.

Topic 5.61.6% of the exam4 of 7 practice exams (6 questions)6 FRQs since 2012

Try it

The graph of

f'

, the derivative of

f

, on

0 < x < 8

is shown.

Where is the graph of

f

concave down, and where does it have points of inflection?

Solution

concave down on $(2, 5)$ , where $f'$ falls, with points of inflection at $x = 2$ and $x = 5$ . That $f'$ is positive says only that $f$ is increasing.

Common mistake

reading concavity from the sign of

f'

(“

f' > 0

, so concave up”) instead of from whether

f'

is increasing or decreasing.

Practice: Reading f From the Graph of f′ worksheet · AP Unit 5 worksheet

2. Absolute extrema from the graph of $f'$ and areas

Start from the value you are given and add the signed areas between the graph of $f'$ and the axis. The candidates are the endpoints and the points where $f'$ changes sign.

Topic 5.51.5% of the exam2 of 7 practice exams (2 questions)9 FRQs since 2012

Try it

The graph of

f'

0 \le x \le 6

is shown. It is made of line segments, and the regions between the graph and the

x

-axis have areas 3 and 5.

f(0) = 4

, what are the absolute maximum and minimum values of

f

[0, 6]

Solution

$f(2) = 4 + 3 = 7$ and $f(6) = 7 - 5 = 2$ . Comparing 4, 7 and 2, the maximum is 7 and the minimum is 2.

Common mistake

giving the

x

-value where the extremum occurs when the question asks for the value, or forgetting to start from the given value

f(0)

Practice: Reading f From the Graph of f′ worksheet · AP Unit 5 worksheet

3. Absolute extrema on a closed interval (the candidates test)

Evaluate $f$ at the critical points inside the interval and at both endpoints. The largest value is the absolute maximum, the smallest the absolute minimum.

Topic 5.51.3% of the exam3 of 7 practice exams (4 questions)2 FRQs since 2012

Try it

[-2, 6]

, what is the largest value that

f(x) = x^3 - 3x^2 - 9x + 5

takes?

Solution

$f'(x) = 3(x - 3)(x + 1)$ . The candidates are $f(-2) = 3$ , $f(-1) = 10$ , $f(3) = -22$ and $f(6) = 59$ , so the absolute maximum is 59, at the endpoint $x = 6$ .

Common mistake

skipping the endpoints (here that gives 10, the relative maximum), or giving the

x

-value instead of the value.

Practice: Absolute Extrema on a Closed Interval worksheet · AP Unit 5 worksheet

4. Increasing and decreasing intervals

$f$ is increasing where $f' > 0$ and decreasing where $f' < 0$ . Factor $f'$ and make a sign chart.

Topic 5.31.2% of the exam5 of 7 practice exams (6 questions)2 FRQs since 2012

Try it

On what interval is

f(x) = xe^{-x}

increasing?

Solution

$f'(x) = (1 - x)e^{-x}$ . Since $e^{-x}$ is always positive, the sign is the sign of $1 - x$ , and $f$ is increasing on $(-\infty, 1)$ .

Common mistake

reading the intervals off the sign of

f''

or of

f

itself, or letting a factor that is always positive (such as

e^{-x}

) change the sign chart.

Practice: Increasing and Decreasing Intervals worksheet · AP Unit 5 worksheet

5. Relative extrema from a formula for $f'$

$f$ has a relative maximum where $f'$ changes from positive to negative, and a relative minimum where it changes from negative to positive. A zero of $f'$ with no sign change is neither.

Topic 5.41.1% of the exam6 of 7 practice exams (7 questions)2 FRQs since 2012

Try it

f'(x) = (x - 1)(x - 4)^2(x + 2)

. Where does

f

have relative extrema?

Solution

$f'$ goes from positive to negative at $x = -2$ (relative maximum) and from negative to positive at $x = 1$ (relative minimum). At $x = 4$ the squared factor keeps $f'$ positive on both sides, so there is no extremum there.

Common mistake

counting every zero of

f'

as an extremum. On the calculator section the derivative is often something like

\cos(x^2)

, and you find its sign changes from a graph.

Practice: Relative Extrema worksheet · AP Unit 5 worksheet

6. Points of inflection from a formula

A point of inflection is where $f''$ changes sign, not just where it is 0.

Topic 5.61.0% of the exam5 of 7 practice exams (5 questions)no FRQs since 2012 (the last was in 2008)

Try it

Where does the graph of

f(x) = x^5 - 5x^4

have points of inflection?

Solution

$f''(x) = 20x^2(x - 3)$ . It is 0 at $x = 0$ and $x = 3$ , but changes sign only at $x = 3$ , so that is the only point of inflection.

Common mistake

calling every zero of

f''

a point of inflection (here

x = 0

), or using the zeros of

f'

, which are critical points.

Practice: Concavity and Inflection Points worksheet · AP Unit 5 worksheet

7. Concavity intervals from a formula

The graph is concave up exactly where $f'' > 0$ and concave down where $f'' < 0$ .

Topic 5.61.0% of the exam5 of 7 practice exams (6 questions)3 FRQs since 2012, each as part of a larger question

Try it

On what interval is the graph of

f(x) = xe^x

concave up?

Solution

$f'(x) = (x + 1)e^x$ and $f''(x) = (x + 2)e^x$ , which is positive for $x > -2$ . Concave up on $(-2, \infty)$ .

Common mistake

using the sign of

f'

(that gives

(-1, \infty)

, where

f

is increasing), or flipping the sign at a squared factor of

f''

Practice: Concavity and Inflection Points worksheet · AP Unit 5 worksheet

8. Mean Value Theorem: find $c$ or bound a value

If $f$ is differentiable on $(a, b)$ and continuous on $[a, b]$ , some $c$ in between has $f'(c) = \frac{f(b) - f(a)}{b - a}$ .

Topic 5.10.9% of the exam5 of 7 practice exams (5 questions)1 FRQ since 2012 (2025)

Try it

For

f(x) = x^3

[0, 3]

, which

c

does the Mean Value Theorem promise?

Solution

the average rate of change is $\frac{27 - 0}{3} = 9$ . Solving $3c^2 = 9$ gives $c = \sqrt{3}$ , which is in $(0, 3)$ .

Common mistake

taking the midpoint of the interval (1.5) for

c

, or giving the average rate of change (9) as the answer.

Practice: The Mean Value Theorem worksheet · AP Unit 5 worksheet

9. Which theorem applies (MVT, Rolle, EVT, IVT)?

Match the conclusion to the theorem and check its hypotheses. The IVT and the EVT are about values of $f$ ; the MVT and Rolle’s theorem are about $f'$ and need differentiability.

Topic 5.10.9% of the exam4 of 7 practice exams (5 questions)no FRQs since 2012 (the last was in 2011)

Try it

f

is differentiable for all

x

, with

f(0) = 1

and

f(4) = 9

. Must there be a

c

(0, 4)

with

f'(c) = 2

? With

f(c) = 5

Solution

yes to both. The average rate of change is $\frac{9 - 1}{4} = 2$ , so the MVT gives $f'(c) = 2$ . And $f$ is continuous with 5 between 1 and 9, so the IVT gives $f(c) = 5$ .

Common mistake

using a theorem whose hypotheses are not given, such as the MVT for a function that is only known to be continuous, or the IVT to conclude something about

f'

Practice: The Mean Value Theorem worksheet · AP Unit 5 worksheet

10. Increasing intervals and extrema from the graph of $f'$

Read the sign of $f'$ off the graph: $f$ increases where the graph of $f'$ is above the axis and has a relative maximum where it crosses from above to below.

Topic 5.40.9% of the exam2 of 7 practice exams (2 questions)7 FRQs since 2012

Try it

The graph of

f'

is shown. It meets the

x

-axis at

x = 1

x = 3

and

x = 5

At each of these, does

f

have a relative maximum, a relative minimum or neither?

Solution

relative maximum at $x = 1$ , relative minimum at $x = 5$ , and neither at $x = 3$ , where $f'$ is negative on both sides.

Common mistake

reading the behavior of

f

off the shape of the graph of

f'

: a peak of

f'

, or a point where it only touches the axis, is not an extremum of

f

. On free response, the reason “

f'

changes from positive to negative” earns the point.

Practice: Reading f From the Graph of f′ worksheet · AP Unit 5 worksheet

11. Horizontal and vertical tangents on an implicit curve

Write $\frac{dy}{dx}$ as a fraction. The tangent is horizontal where the numerator is 0 (and the denominator is not), vertical where the denominator is 0. Then find the points that are on the curve.

Topic 5.120.9% of the examno multiple choice questions5 FRQs since 2012 (2015, 2021, 2023, 2024, 2025)

Try it

At which points of the curve

x^2 + xy + y^2 = 27

is the tangent line horizontal?

Solution

$\frac{dy}{dx} = -\frac{2x + y}{x + 2y}$ , so $y = -2x$ . Substituting, $3x^2 = 27$ and $x = \pm 3$ : the points are $(3, -6)$ and $(-3, 6)$ .

Common mistake

swapping the conditions (setting the denominator to 0 for a horizontal tangent), or stopping at

y = -2x

without finding the points on the curve.

Practice: Implicit Differentiation worksheet · AP Unit 5 worksheet

12. The second derivative of an implicit relation

Differentiate $\frac{dy}{dx}$ again, treating $y$ as a function of $x$ , then substitute the value of $\frac{dy}{dx}$ at the point.

Topic 5.120.8% of the exam1 of 7 practice exams (1 question)7 FRQs since 2012

Try it

For

x^2 + y^2 = 25

, find

\frac{d^2y}{dx^2}

(3, 4)

Solution

$\frac{dy}{dx} = -\frac{x}{y} = -\frac{3}{4}$ . Then $\frac{d^2y}{dx^2} = -\frac{y - x \cdot \frac{dy}{dx}}{y^2} = -\frac{4 + \frac{9}{4}}{16} = -\frac{25}{64}$ .

Common mistake

treating

y

as a constant when differentiating

\frac{dy}{dx}

again. Every

y

brings in another

\frac{dy}{dx}

Practice: Implicit Differentiation worksheet · AP Unit 5 worksheet

Nine more Unit 5 types, lower priority

13. Justify with the Mean Value Theorem (free response)

A full justification names the hypothesis, shows the computation and states the conclusion.

Topic 5.10.8% of the examno multiple choice questions7 FRQs since 2012, 2 of them on the calculator section

Try it

r

is differentiable,

r(2) = 10

and

r(8) = 10

. Must there be a

c

(2, 8)

with

r'(c) = 0

Solution

yes. $r$ is differentiable, so it is continuous on $[2, 8]$ , and $\frac{r(8) - r(2)}{8 - 2} = 0$ . By the Mean Value Theorem, $r'(c) = 0$ for some $c$ in $(2, 8)$ .

Common mistake

writing “by the MVT” without the hypothesis and the computation, or citing the IVT for a conclusion about

r'

Practice: The Mean Value Theorem worksheet · AP Unit 5 worksheet

14. Connect the signs of $f$ , $f'$ and $f''$

$f' > 0$ means $f$ rises. $f'' > 0$ means $f'$ rises and the graph bends up, staying above its tangent lines.

Topic 5.90.7% of the exam3 of 7 practice exams (5 questions)no FRQs since 2012 (the last was in 2009)

Try it

f'(x) > 0

and

f''(x) < 0

for all

x

f(1) = 3

and

f'(1) = 1

. Could

f(2)

be 4.5?

Solution

no. $f$ is increasing, so $f(2) > 3$ , and concave down, so $f(2)$ is below the tangent line value $3 + 1 = 4$ . So $3 < f(2) < 4$ .

Common mistake

reading the sign of

f'

for whether

f'

is increasing:

f'

is positive where

f

rises and increasing where the graph is concave up.

Practice: Curve Sketching worksheet · AP Unit 5 worksheet

15. Match the graphs of $f$ and $f'$

The graph of $f'$ is above the axis where $f$ rises, below where it falls, and crosses 0 at every horizontal tangent of $f$ .

Topic 5.80.7% of the exam5 of 7 practice exams (5 questions)never asked on the free response

Try it

The graph of

f(x) = x^3 - 3x

is shown, with horizontal tangents at

x = -1

and

x = 1

What does the graph of

f'

look like?

Solution

an upward parabola, positive outside $[-1, 1]$ and negative inside, crossing the axis at $x = -1$ and $x = 1$ : $f'(x) = 3x^2 - 3$ .

Common mistake

matching the shape of

f

instead of its slopes, such as picking a graph with peaks where

f

has peaks.

Practice: Graphing the Derivative worksheet · AP Unit 5 worksheet

16. Optimization

Write the quantity in one variable using the constraint, find its critical points on the allowed domain, and check that you have a maximum or minimum.

Topic 5.110.6% of the exam1 of 7 practice exams (1 question)no FRQs since 2012 (the last was in 2009)

Try it

An open-top box with a square base must hold 32 cubic inches. What dimensions use the least material?

Solution

with base side $x$ , the height is $\frac{32}{x^2}$ and the area is $S = x^2 + \frac{128}{x}$ . $S' = 2x - \frac{128}{x^2} = 0$ gives $x = 4$ , and $S'' > 0$ there. The box is 4 by 4 by 2 inches, using 48 square inches.

Common mistake

stopping at the critical point instead of giving the quantity asked for, or not checking that it is a minimum.

Practice: Optimization worksheet · AP Unit 5 worksheet

17. Critical points

A critical point is in the domain of $f$ where $f' = 0$ or $f'$ does not exist.

Topic 5.20.5% of the exam2 of 7 practice exams (2 questions)2 FRQs since 2012, each as part of a larger question

Try it

Find the critical points of

f(x) = x^{1/3}(x - 4)

Solution

$f'(x) = \frac{4x - 4}{3x^{2/3}}$ . It is 0 at $x = 1$ and undefined at $x = 0$ , which is in the domain, so both are critical points.

Common mistake

leaving out the point where

f'

does not exist (

x = 0

), or solving

f(x) = 0

instead.

Practice: Critical Points worksheet · AP Unit 5 worksheet

18. Relative extrema from a formula for $f$

Differentiate, find the critical points, and classify them with the sign of $f'$ or the second derivative test.

Topic 5.40.5% of the examnot on the 2012–2018 practice exams (the last was in 2008)no FRQs since 2012

Try it

Find the relative maximum value of

f(x) = x^3 - 12x + 1

Solution

$f'(x) = 3(x - 2)(x + 2)$ , which changes from positive to negative at $x = -2$ . The relative maximum value is $f(-2) = 17$ (the relative minimum is $f(2) = -15$ ).

Common mistake

solving

f(x) = 0

instead of

f'(x) = 0

, or naming the other kind of extremum.

Practice: Relative Extrema worksheet · AP Unit 5 worksheet

19. Which table fits the signs of $f'$ and $f''$ ?

With equally spaced inputs, $F' > 0$ means the values go up; $F'' > 0$ means the differences between them grow too.

Topic 5.90.3% of the exam3 of 7 practice exams (3 questions)never asked on the free response

Try it

F' > 0

and

F'' > 0

. Which table of values of

F

fits?

$x$	0	1	2	3
Table A: $F(x)$	1	2	4	8
Table B: $F(x)$	1	4	6	7

Solution

Table A. Its differences 1, 2, 4 grow; the differences of the other, 3, 2, 1, shrink, which fits a concave-down function.

Common mistake

checking only that the values increase.

Practice: Curve Sketching worksheet · AP Unit 5 worksheet

20. The second derivative test

If $f'(c) = 0$ , then $f''(c) < 0$ gives a relative maximum and $f''(c) > 0$ a relative minimum. If $f''(c) = 0$ , the test decides nothing.

Topic 5.70.3% of the examno multiple choice questions2 FRQs since 2012 (2015, 2021)

Try it

f'(2) = 0

and

f''(2) = -5

. What happens at

x = 2

? And if

f'(0) = 0

and

f''(0) = 0

Solution

a relative maximum at $x = 2$ (concave down). At $x = 0$ the test is inconclusive: $f(x) = x^4$ has those values and a minimum there, while $x^3$ has them and no extremum.

Common mistake

reading the test backwards (

f'' < 0

as a minimum), or using it where

f'(c) \ne 0

Practice: Relative Extrema worksheet · AP Unit 5 worksheet

21. Sketch the graph of $f$ or $f'$ (free response)

Relative extrema come from the sign changes of $f'$ , points of inflection from the sign changes of $f''$ .

Topic 5.80.3% of the examno multiple choice questionsno FRQs since 2012 (the last was in 2005)

Try it

f'(x) = (x - 1)(x - 4)

. Where should a sketch of

f

show its turning points and its point of inflection?

Solution

a relative maximum at $x = 1$ , a relative minimum at $x = 4$ , and a point of inflection at $x = \frac{5}{2}$ , where $f''(x) = 2x - 5$ changes sign.

Common mistake

drawing the extrema where

f''

changes sign, or the inflection points where

f'

does.

Practice: Curve Sketching worksheet · AP Unit 5 worksheet

What’s rising and fading in Unit 5

We compare each type’s share of recent exams (multiple choice 2012–2018, free response 2012–2026) with its share across all years: Rising is at least double, Fading under half. No Unit 5 type is rising by that measure. The movement is toward reading graphs and implicit curves, and away from the classic word problems.

Up: the graph of $f'$ . Concavity from the graph of $f'$ is 2.13% of recent exams against 1.28% across all years, and absolute extrema from the graph of $f'$ 1.87% against 1.07%, the two biggest gains in the unit. The second went from 2.4% of the free-response points (2012–2018) to 3.8% (2019–2026).
Up on the newest free response: horizontal and vertical tangents. 0.8% of the free-response points in 2012–2018, 3.4% in 2019–2026, the largest move of any Unit 5 type, with questions in 2021, 2023, 2024 and 2025.
Fading: optimization. 0.16% of recent exams against 1.37% across all years. Its last official multiple choice question was in 2016 and its last free-response question in 2009. Teacher material confirms the drop.
Fading: critical points (0.32% against 0.74%), and the candidates test from a formula (0.96% against 2.54%), which teachers still drill heavily.
Dormant: relative extrema from a formula for $f$ (last official multiple choice question in 2008) and sketching a graph (last free-response question in 2005).

Unit 5 as a whole went from 13.0% to 17.0% of the free-response points. One question is about 2.3% of a window’s points, so read each move as a question or two.

In the CED, rarely asked in multiple choice

The second derivative test (topic 5.7). Not one official multiple choice question, but 7 free-response questions since 1998.
Implicit relations (topic 5.12). Two official multiple choice questions, both on the second derivative, the last in 2012. Free response asked the second derivative in 13 questions since 1998 and horizontal or vertical tangents in 10.
Introduction to optimization (topic 5.10) has never been the main topic of an official question, and solving optimization problems (5.11) has 5 official multiple choice questions, only 1 from 2012–2018.
Justifying with the Mean Value Theorem has never been a multiple choice question, but it was in 19 free-response questions since 1998.

So a Unit 5 study plan built only from multiple choice misses implicit curves and MVT justifications, which carry free-response points almost every year.

What students over- and under-practice in Unit 5

We tagged 10,263 multiple choice questions from teacher worksheets, review packets and prep books (reposted official questions and non-AB content removed). Unit 5 is 16.8% of them, against 19.9% of the 719 in-scope official multiple choice questions from 1969–2018. Per type:

Question type	Practice vs official share	Ratio
Mean Value Theorem: find c or bound	1.73% vs 0.83%	2.1× over
Optimization	1.15% vs 0.70%	1.6× over
Second derivative of an implicit relation	0.45% vs 0.28%	1.6× over
Critical points	0.74% vs 0.56%	1.3× over
Relative extrema from a formula for f	0.97% vs 0.83%	1.2× over
Candidates test	1.43% vs 1.39%	1.0×, even
Signs of f, f′, f″	1.42% vs 1.39%	1.0×, even
Concavity from the graph of f′	0.92% vs 1.11%	0.8× under
Extrema from the graph of f′	0.91% vs 1.11%	0.8× under
Points of inflection from a formula	1.48% vs 1.95%	0.8× under
Match the graphs of f and f′	0.95% vs 1.53%	0.6× under
Which theorem applies	1.11% vs 1.81%	0.6× under
Absolute extrema from the graph of f′	0.24% vs 0.42%	0.6× under
Increasing and decreasing intervals	1.10% vs 2.09%	0.5× under
Concavity intervals from a formula	0.89% vs 1.67%	0.5× under
Relative extrema from a formula for f′	0.59% vs 1.53%	0.4× under
Which table fits the signs	0.06% vs 0.70%	0.1× under

Official counts per type are small (2 to 15 questions), so read the ratios as directions.

Practice leans on the fading word problems. Optimization gets 1.6 times its official share, on a type that has not been on the free response since 2009.
Mean Value Theorem calculations are over-practiced; MVT justifications are not practiced at all, because there is no multiple choice version to copy. The free response asks the justification far more often.
The biggest gaps are the calculator-section staples: relative extrema from a formula for $f'$ (0.4×) and the table that fits the signs of $f'$ and $f''$ (6 practice questions in our whole set).
Graphs of $f'$ are under-practiced in every form, though they are the fastest-growing part of the unit.

Calculator note

Unit 5 is split almost evenly. Of the 65 Unit 5 multiple choice questions on the 2012–2018 practice exams, 33 were in the no-calculator part and 32 in the calculator part. The calculator part had 6 of the 7 relative-extrema-from- $f'$ questions, 5 of the 6 concavity-from-the-graph questions, all 3 table questions and both increasing-from-the-graph questions. Increasing intervals (5 of 6), points of inflection (4 of 5), the Mean Value Theorem (4 of 5) and the candidates test (3 of 4) were mostly no-calculator.

Across all 17 official exams, 62 Unit 5 multiple choice questions were in the calculator section. 25 of them work from a graph or a table, where the calculator is allowed but rarely needed. 29 give a formula, often one you cannot solve by hand, such as $f'(x) = \cos(x^2)$ , and ask where it changes sign; relative extrema from a formula for $f'$ had 9 of its 11 official questions there.

On free response, only 5 of the 34 questions with a Unit 5 part allowed a calculator (counting 2020, when a calculator was allowed throughout): 3 of the 8 Mean Value Theorem questions, 1 of the 11 graph-of- $f'$ questions and none of the 10 implicit-curve questions.

So know how to graph $f'$ and find its zeros on your calculator, and give numerical answers to three decimal places. Do the graph reading, the implicit curves and the justifications by hand.

How we counted

Every number here comes from the public official AP Calculus AB exams: 760 multiple choice questions (1969–2018) and 416 free-response questions (1969–2026), each tagged to its CED topic and question type. Every example problem is our own.

Practice Unit 5

Our free AP Calculus AB Unit 5 worksheet has 20 original multiple choice questions on extrema, concavity, the Mean Value Theorem and the graphs of $f$ , $f'$ and $f''$ , one on each of the first 19 types above and one on setting up an optimization problem (topic 5.10), the first step of every optimization question, in place of the second derivative test (sketching a graph is left out too). Part A has 12 questions without a calculator and Part B has 8 with one. Most Part B questions are graphs of $f'$ , matching graphs, tables and which theorem applies, where the calculator does not help. Two in each version need a graphing calculator, and numerical answers are given to three decimal places. It comes with an answer key and a printable PDF.

For free response, the AP Calculus AB Unit 5 FRQ set has 4 multi-part questions in the exam’s form, each scored out of 9 points, with a scoring guide and a PDF. Both make good practice before the Unit 5 Progress Check in AP Classroom.

Common Questions

The CED gives Unit 5 15–20% of the multiple choice. On the 7 official practice exams from 2012–2018 it was 65 of 315 multiple choice questions (20.6%), 8 to 12 per exam, more than any other unit. In free response it earned 15.0% of the points on the 2012–2026 exams, with a Unit 5 part on every exam: the graph of f′, implicit curves, Mean Value Theorem justifications and the candidates test.

Often, and more than before. A free-response question built on the graph of f′ was on 11 of the 15 exams from 2012 to 2026. It asks where f is increasing (f′ above the axis), where f has a relative maximum (f′ changes from positive to negative), where the graph of f is concave up (f′ increasing) and the absolute maximum of f, found from a given value plus the areas between the graph and the axis.

State that the function is differentiable (so it is continuous) on the interval, compute the average rate of change from the given values, and conclude that the derivative equals that value somewhere in the open interval. The free response asked an MVT justification in 19 questions since 1998, usually from a table. There is no multiple choice version, so practice writing it.

Yes, mostly on the free response. Implicit-curve questions were on 7 of the 15 exams from 2012 to 2026: the second derivative at a point, and where the tangent line is horizontal (numerator of dy/dx is 0) or vertical (denominator is 0). Horizontal and vertical tangents were asked in 2021, 2023, 2024 and 2025, all without a calculator.

Sometimes. Of the 65 Unit 5 multiple choice questions on the 2012–2018 practice exams, 32 were in the calculator part. Graph and table questions there rarely need it; it is needed when f′ is a function you cannot solve by hand, such as cos(x²): graph it, find where it changes sign, and give a numerical answer to three decimal places. Relative extrema from a formula for f′ is the classic: 9 of its 11 official multiple choice questions were in the calculator section. Only 5 of the 34 Unit 5 free-response questions since 2012 allowed a calculator.

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AP Calculus AB Unit 5 Review: The Most-Tested Extrema and Concavity Questions

The 12 most-tested Unit 5 question types

1. Concavity and inflection from the graph of f′f'f′

2. Absolute extrema from the graph of f′f'f′ and areas

3. Absolute extrema on a closed interval (the candidates test)

4. Increasing and decreasing intervals

5. Relative extrema from a formula for f′f'f′

6. Points of inflection from a formula

7. Concavity intervals from a formula

8. Mean Value Theorem: find ccc or bound a value

9. Which theorem applies (MVT, Rolle, EVT, IVT)?

10. Increasing intervals and extrema from the graph of f′f'f′

11. Horizontal and vertical tangents on an implicit curve

12. The second derivative of an implicit relation

Nine more Unit 5 types, lower priority

13. Justify with the Mean Value Theorem (free response)

14. Connect the signs of fff, f′f'f′ and f′′f''f′′

15. Match the graphs of fff and f′f'f′

16. Optimization

17. Critical points

18. Relative extrema from a formula for fff

19. Which table fits the signs of f′f'f′ and f′′f''f′′?

20. The second derivative test

21. Sketch the graph of fff or f′f'f′ (free response)

What’s rising and fading in Unit 5

In the CED, rarely asked in multiple choice

What students over- and under-practice in Unit 5

Calculator note

How we counted

Practice Unit 5

Common Questions

How much of the AP Calculus AB exam is Unit 5?+

How is the graph of f′ tested on the AP Calculus AB exam?+

How do I justify an answer with the Mean Value Theorem?+

Are implicit curves on the AP Calculus AB exam?+

Do I need a calculator for Unit 5 questions?+

Practice this unit: free AP Calculus AB Unit 5 worksheet

Keep reading

AP Calculus AB Review: The Most-Tested Questions, Unit by Unit

AP Calculus AB Unit 1 Review: The Most-Tested Limits and Continuity Questions

AP Calculus AB Unit 2 Review: The Most-Tested Derivative Questions

1. Concavity and inflection from the graph of $f'$

2. Absolute extrema from the graph of $f'$ and areas

5. Relative extrema from a formula for $f'$

8. Mean Value Theorem: find $c$ or bound a value

10. Increasing intervals and extrema from the graph of $f'$

14. Connect the signs of $f$ , $f'$ and $f''$

15. Match the graphs of $f$ and $f'$

18. Relative extrema from a formula for $f$

19. Which table fits the signs of $f'$ and $f''$ ?

21. Sketch the graph of $f$ or $f'$ (free response)