Unit 5Evaluating f at Critical Points and Endpoints

Concept 4.146Built for May 2027 AP Exam

Evaluating f at Critical Points and Endpoints

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

Once the candidates are listed, the rest is evaluation, and evaluation is where answers quietly go wrong. The work is to compute the function at each candidate, arrange the results so they can be compared, and report the winning value with the input that produced it. Most of the errors here are not conceptual: a value is dropped, an endpoint is evaluated with the wrong expression, or a rate model is evaluated where the accumulated total was needed. A small table of candidates and values prevents all three, because it makes the omissions visible before the comparison rather than after.

On the AP Exam

What you’ll see: A table of rates with an accumulation function, or a formula with several candidates, followed by a request for the absolute maximum or minimum value on a closed interval.
What you’ll do: Evaluate the accumulation or the function at every candidate, including the endpoints, then compare the numbers and report the extreme with its location.
Memory hook: Compute all, compare once. Every candidate gets a number before any candidate gets a verdict.
Use it on the exam: For MCQs: The distractors are values at the wrong candidate or at a critical number outside the interval, so a complete table eliminates them quickly. For FRQs: On 2019 Q1(c) the answer depends on accumulated totals rather than on the rate values themselves, and the guideline awards the point for comparing those totals at the correct times.

Concept Review

The Table as a Habit

Figure 8 — A candidate list with values attached. The two shaded rows are the winners, and the shading is the only judgement the table makes.

The table has one row per candidate and three columns of information: the input, the reason it is a candidate, and the output. Building it in that order means the list is complete before any arithmetic starts, and completeness is the property most responses lose. A comparison written as a sentence like "comparing $3.1$ , $8.4$ , $2.6$ , and $9.7$ " is harder to check than the same numbers in a column, and the reader is checking.

A four-row evaluation table listing two endpoints and two critical numbers with their function values, the largest row highlighted in amber and the smallest in blue

When the Function Is an Accumulation

If $A(x)=\int_{a}^{x} r(t)\,dt$ , then $A'(x)=r(x)$ , so the critical numbers of $A$ are the places where the rate is zero or the rate is undefined. The evaluation step then needs accumulated totals, not rate values. This is the single most common mix-up in this topic: the table gives $r$ , the question asks about $A$ , and the answer requires adding before comparing. A response that compares the rates has compared the wrong function.

Sign, Units, and Reporting

Three presentation details carry points:

The location goes with the value. "The maximum is $9.7$ at $x=b$ " is complete; " $9.7$ " is not.
The units belong to the output only. A rate in gallons per second integrates to gallons, and the extreme of an accumulation is measured in the accumulated quantity.
A tie must be reported as a tie. When two candidates produce the same extreme value, both locations belong in the answer, and naming one is incomplete rather than wrong.

Comparing Without a Formula

When the function is known only at selected points, the comparison still works, but it rests on information the question supplies. The candidate list comes from the stated behaviour of the derivative, and the values come from the table. The theorem in 5.3 is what licenses the claim that no other point can beat the candidates, so a table-only answer still needs the continuity sentence to be complete.

Reading the Arithmetic Backwards

A useful check costs a few seconds: substitute the winning input into the original expression and confirm the output, and confirm that the winning output really is the largest in the column. Reversing the roles of input and output is a surprisingly common slip when the table lists inputs in one order and values in another, and the check catches it immediately.

Common Misconception Check

Myth 1: "The largest number in the table is the maximum." Truth: The largest number in the column of outputs is the maximum. Inputs and outputs sit in different columns, and confusing them produces an answer with the wrong kind of quantity.
Myth 2: "The rate and the accumulated total peak at the same input." Truth: They do not. The accumulation peaks where its rate is zero, which is generally not where the rate is largest.
Myth 3: "One location is enough when two candidates tie." Truth: A tie means both inputs produce the extreme, and the answer must name both.

Topic-Specific Study Method

Build the Column Before the Comparison

Borrowed idea: The table is a promise that nothing was skipped, and a promise is worth more than speed. Fill the rows in the order of the candidates, from left endpoint to right endpoint, and leave the last column empty until the end. Then compare. Then write one sentence per extreme with the value and the location together, which is the shape the reader is looking for and the shape that survives a second reading. So what do you do when the function is only known through a rate? Convert the rate into totals first, and only then compare.

Function type	What to evaluate	What not to compare
explicit formula	the formula at each candidate	derivative values
accumulation $\int r$	running totals at each candidate	the rate values
table only	the listed values at the candidates	interpolated guesses
model with units	the output with its unit stated	inputs as if they were outputs

① Fill in the endpoint rows first. They are the rows most often left blank, and leaving them blank is the error the table exists to prevent.

② Keep full precision until the comparison, then round. Rounding early can turn a tie into a near-tie and produce a single location where two belong.

③ Write the unit once, at the end, next to the value. Repeating the unit inside the comparison column adds noise without adding information.

Training & Error Diagnosis

Core Archetype (Simplest Form): Find the absolute maximum of $f(x)=x^{3}-3x$ on $[-2,2]$ .

Standard Execution:

Candidates. $f'(x)=3(x-1)(x+1)$ gives $x=\pm 1$ , and the endpoints are $\pm 2$ .

Evaluate. $f(-2)=-2$ , $f(-1)=2$ , $f(1)=-2$ , $f(2)=2$ .

Compare. The largest output is $2$ , produced twice.

Report. The absolute maximum is $2$ , attained at $x=-1$ and at $x=2$ .

Exam Variation (Context / Parameter Shift): Water enters a tank at a rate $r(t)$ , and the volume is $V(x)=\int_{0}^{x} r(t)\,dt$ . The rate is zero at $t=4$ and $t=9$ , and the accumulated volumes are $V(0)=0$ , $V(4)=18$ , $V(9)=11$ , and $V(12)=20$ . Find the maximum volume on $[0,12]$ .

Solution: The candidates are $0$ , $4$ , $9$ , and $12$ , with volumes $0$ , $18$ , $11$ , and $20$ . The maximum volume is $20$ at $t=12$ , which is larger than the local peak of $18$ at $t=4$ . The comparison is between accumulated volumes, not between rates, and the endpoint wins.

Common Traps & Failure Signals:

The rate compared (Signal: the largest rate offered as the maximum volume): the wrong function was tabulated.
The blank endpoint (Signal: a comparison of interior candidates only): an endpoint row was never filled.
The single location (Signal: one input reported for a tied extreme): a tie was reported as if it were unique.

6D Cognitive Error Diagnosis (Where Your Thinking Broke):

Column Empty: You compared before you evaluated.
Function Swapped: You tabulated the integrand instead of the integral, which is the accumulation version of answering a different question with correct arithmetic.
Tie Collapsed: You found the extreme value and stopped at the first input that produced it, which loses the second location on a part whose guideline does score both locations separately.

Past 10-Year AP Exam Appearance

Testing Frequency: Nearly Every Year; Seven released BC sets make the evaluation step the substance of the part, and in every one of them the function is supplied as a table of rates or as the graph of its own derivative rather than as a formula
10-Year Total Count: 7 of 9 Exams · 9 FRQ Parts; Counted where the scored work is converting supplied data into values at the candidates and comparing them
FRQ Years Appeared: 2017 (Q3c), 2018 (Q1d), 2019 (Q1c, Q3c), 2022 (Q1d), 2023 (Q4d), 2025 (Q1D, Q4D), 2026 (Q4D)

Practice Examples

MCQ 1

Four Values

A differentiable function has candidates $x=1$ , $x=3$ , $x=6$ , and $x=8$ on its interval, with values $f(1)=5$ , $f(3)=12$ , $f(6)=-2$ , and $f(8)=12$ . What is the absolute maximum?

MCQ 2

An Accumulation

A function is defined by $V(x)=\int_{0}^{x} r(t)\,dt$ with $V(0)=0$ , $V(2)=9$ , $V(5)=4$ , and $V(7)=6$ . On $[0,7]$ , where does $V$ attain its maximum?

MCQ 3

A Rate Trap

The rate $r(t)$ in gallons per minute is largest at $t=5$ . The accumulated volume is $V(x)=\int_{0}^{x} r(t)\,dt$ . Which statement must be true?

FRQ 1 • From Rates to Totals

Water flows into a tank at a rate $r(t)$ given in the table, where $t$ is measured in minutes and $r(t)$ in liters per minute. Let $W(x)=\int_{0}^{x} r(t)\,dt$ with $W(0)=0$ .

$t$	$0$	$3$	$6$	$9$	$12$
$r(t)$	$2$	$8$	$5$	$0$	$6$

(a) Use a left Riemann sum with the four subintervals to approximate $W(12)$ .

(b) It is known that $W(3)=15$ , $W(6)=34$ , and $W(9)=40$ . Explain why $W$ has a critical number at $t=9$ .

(c) Find the maximum value of $W$ on $[0,12]$ , using $W(9)=40$ and the approximation of $W(12)$ from part (a).

(d) A student compares the values of $r(t)$ and concludes that $W$ is largest at $t=3$ . Explain the error.

Cheatsheet

Evaluating Candidates

	What to compute	Common error
explicit formula	$f$ at each candidate	comparing $f'$ values
accumulation	running totals	comparing rates
table only	the listed values	interpolating
tie	both locations	one location
report	value plus location plus unit	value alone

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Unit 5Evaluating f at Critical Points and Endpoints

Concept 4.146Built for May 2027 AP Exam

Evaluating f at Critical Points and Endpoints

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

On the AP Exam

What you’ll see: A table of rates with an accumulation function, or a formula with several candidates, followed by a request for the absolute maximum or minimum value on a closed interval.
What you’ll do: Evaluate the accumulation or the function at every candidate, including the endpoints, then compare the numbers and report the extreme with its location.
Memory hook: Compute all, compare once. Every candidate gets a number before any candidate gets a verdict.
Use it on the exam: For MCQs: The distractors are values at the wrong candidate or at a critical number outside the interval, so a complete table eliminates them quickly. For FRQs: On 2019 Q1(c) the answer depends on accumulated totals rather than on the rate values themselves, and the guideline awards the point for comparing those totals at the correct times.

Concept Review

The Table as a Habit

Figure 8 — A candidate list with values attached. The two shaded rows are the winners, and the shading is the only judgement the table makes.

A four-row evaluation table listing two endpoints and two critical numbers with their function values, the largest row highlighted in amber and the smallest in blue

When the Function Is an Accumulation

Sign, Units, and Reporting

Three presentation details carry points:

The location goes with the value. "The maximum is $9.7$ at $x=b$ " is complete; " $9.7$ " is not.
The units belong to the output only. A rate in gallons per second integrates to gallons, and the extreme of an accumulation is measured in the accumulated quantity.
A tie must be reported as a tie. When two candidates produce the same extreme value, both locations belong in the answer, and naming one is incomplete rather than wrong.

Comparing Without a Formula

Reading the Arithmetic Backwards

Common Misconception Check

Myth 1: "The largest number in the table is the maximum." Truth: The largest number in the column of outputs is the maximum. Inputs and outputs sit in different columns, and confusing them produces an answer with the wrong kind of quantity.
Myth 2: "The rate and the accumulated total peak at the same input." Truth: They do not. The accumulation peaks where its rate is zero, which is generally not where the rate is largest.
Myth 3: "One location is enough when two candidates tie." Truth: A tie means both inputs produce the extreme, and the answer must name both.

Topic-Specific Study Method

Build the Column Before the Comparison

Function type	What to evaluate	What not to compare
explicit formula	the formula at each candidate	derivative values
accumulation $\int r$	running totals at each candidate	the rate values
table only	the listed values at the candidates	interpolated guesses
model with units	the output with its unit stated	inputs as if they were outputs

① Fill in the endpoint rows first. They are the rows most often left blank, and leaving them blank is the error the table exists to prevent.

② Keep full precision until the comparison, then round. Rounding early can turn a tie into a near-tie and produce a single location where two belong.

③ Write the unit once, at the end, next to the value. Repeating the unit inside the comparison column adds noise without adding information.

Training & Error Diagnosis

Core Archetype (Simplest Form): Find the absolute maximum of $f(x)=x^{3}-3x$ on $[-2,2]$ .

Standard Execution:

Candidates. $f'(x)=3(x-1)(x+1)$ gives $x=\pm 1$ , and the endpoints are $\pm 2$ .

Evaluate. $f(-2)=-2$ , $f(-1)=2$ , $f(1)=-2$ , $f(2)=2$ .

Compare. The largest output is $2$ , produced twice.

Report. The absolute maximum is $2$ , attained at $x=-1$ and at $x=2$ .

Solution: The candidates are $0$ , $4$ , $9$ , and $12$ , with volumes $0$ , $18$ , $11$ , and $20$ . The maximum volume is $20$ at $t=12$ , which is larger than the local peak of $18$ at $t=4$ . The comparison is between accumulated volumes, not between rates, and the endpoint wins.

Common Traps & Failure Signals:

The rate compared (Signal: the largest rate offered as the maximum volume): the wrong function was tabulated.
The blank endpoint (Signal: a comparison of interior candidates only): an endpoint row was never filled.
The single location (Signal: one input reported for a tied extreme): a tie was reported as if it were unique.

6D Cognitive Error Diagnosis (Where Your Thinking Broke):

Column Empty: You compared before you evaluated.
Function Swapped: You tabulated the integrand instead of the integral, which is the accumulation version of answering a different question with correct arithmetic.
Tie Collapsed: You found the extreme value and stopped at the first input that produced it, which loses the second location on a part whose guideline does score both locations separately.

Past 10-Year AP Exam Appearance

Testing Frequency: Nearly Every Year; Seven released BC sets make the evaluation step the substance of the part, and in every one of them the function is supplied as a table of rates or as the graph of its own derivative rather than as a formula
10-Year Total Count: 7 of 9 Exams · 9 FRQ Parts; Counted where the scored work is converting supplied data into values at the candidates and comparing them
FRQ Years Appeared: 2017 (Q3c), 2018 (Q1d), 2019 (Q1c, Q3c), 2022 (Q1d), 2023 (Q4d), 2025 (Q1D, Q4D), 2026 (Q4D)

Practice Examples

MCQ 1

Four Values

A differentiable function has candidates $x=1$ , $x=3$ , $x=6$ , and $x=8$ on its interval, with values $f(1)=5$ , $f(3)=12$ , $f(6)=-2$ , and $f(8)=12$ . What is the absolute maximum?

MCQ 2

An Accumulation

A function is defined by $V(x)=\int_{0}^{x} r(t)\,dt$ with $V(0)=0$ , $V(2)=9$ , $V(5)=4$ , and $V(7)=6$ . On $[0,7]$ , where does $V$ attain its maximum?

MCQ 3

A Rate Trap

The rate $r(t)$ in gallons per minute is largest at $t=5$ . The accumulated volume is $V(x)=\int_{0}^{x} r(t)\,dt$ . Which statement must be true?

FRQ 1 • From Rates to Totals

Water flows into a tank at a rate $r(t)$ given in the table, where $t$ is measured in minutes and $r(t)$ in liters per minute. Let $W(x)=\int_{0}^{x} r(t)\,dt$ with $W(0)=0$ .

$t$	$0$	$3$	$6$	$9$	$12$
$r(t)$	$2$	$8$	$5$	$0$	$6$

(a) Use a left Riemann sum with the four subintervals to approximate $W(12)$ .

(b) It is known that $W(3)=15$ , $W(6)=34$ , and $W(9)=40$ . Explain why $W$ has a critical number at $t=9$ .

(c) Find the maximum value of $W$ on $[0,12]$ , using $W(9)=40$ and the approximation of $W(12)$ from part (a).

(d) A student compares the values of $r(t)$ and concludes that $W$ is largest at $t=3$ . Explain the error.

Cheatsheet

Evaluating Candidates

	What to compute	Common error
explicit formula	$f$ at each candidate	comparing $f'$ values
accumulation	running totals	comparing rates
table only	the listed values	interpolating
tie	both locations	one location
report	value plus location plus unit	value alone

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Start Topic Practice Next Topic: 5.9 Concavity Definition

Unit 5Evaluating f at Critical Points and Endpoints

Concept 4.146Built for May 2027 AP Exam

Evaluating f at Critical Points and Endpoints

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

On the AP Exam

What you’ll see: A table of rates with an accumulation function, or a formula with several candidates, followed by a request for the absolute maximum or minimum value on a closed interval.
What you’ll do: Evaluate the accumulation or the function at every candidate, including the endpoints, then compare the numbers and report the extreme with its location.
Memory hook: Compute all, compare once. Every candidate gets a number before any candidate gets a verdict.
Use it on the exam: For MCQs: The distractors are values at the wrong candidate or at a critical number outside the interval, so a complete table eliminates them quickly. For FRQs: On 2019 Q1(c) the answer depends on accumulated totals rather than on the rate values themselves, and the guideline awards the point for comparing those totals at the correct times.

Concept Review

The Table as a Habit

Figure 8 — A candidate list with values attached. The two shaded rows are the winners, and the shading is the only judgement the table makes.

A four-row evaluation table listing two endpoints and two critical numbers with their function values, the largest row highlighted in amber and the smallest in blue

When the Function Is an Accumulation

Sign, Units, and Reporting

Three presentation details carry points:

The location goes with the value. "The maximum is $9.7$ at $x=b$ " is complete; " $9.7$ " is not.
The units belong to the output only. A rate in gallons per second integrates to gallons, and the extreme of an accumulation is measured in the accumulated quantity.
A tie must be reported as a tie. When two candidates produce the same extreme value, both locations belong in the answer, and naming one is incomplete rather than wrong.

Comparing Without a Formula

Reading the Arithmetic Backwards

Common Misconception Check

Myth 1: "The largest number in the table is the maximum." Truth: The largest number in the column of outputs is the maximum. Inputs and outputs sit in different columns, and confusing them produces an answer with the wrong kind of quantity.
Myth 2: "The rate and the accumulated total peak at the same input." Truth: They do not. The accumulation peaks where its rate is zero, which is generally not where the rate is largest.
Myth 3: "One location is enough when two candidates tie." Truth: A tie means both inputs produce the extreme, and the answer must name both.

Topic-Specific Study Method

Build the Column Before the Comparison

Function type	What to evaluate	What not to compare
explicit formula	the formula at each candidate	derivative values
accumulation $\int r$	running totals at each candidate	the rate values
table only	the listed values at the candidates	interpolated guesses
model with units	the output with its unit stated	inputs as if they were outputs

① Fill in the endpoint rows first. They are the rows most often left blank, and leaving them blank is the error the table exists to prevent.

② Keep full precision until the comparison, then round. Rounding early can turn a tie into a near-tie and produce a single location where two belong.

③ Write the unit once, at the end, next to the value. Repeating the unit inside the comparison column adds noise without adding information.

Training & Error Diagnosis

Core Archetype (Simplest Form): Find the absolute maximum of $f(x)=x^{3}-3x$ on $[-2,2]$ .

Standard Execution:

Candidates. $f'(x)=3(x-1)(x+1)$ gives $x=\pm 1$ , and the endpoints are $\pm 2$ .

Evaluate. $f(-2)=-2$ , $f(-1)=2$ , $f(1)=-2$ , $f(2)=2$ .

Compare. The largest output is $2$ , produced twice.

Report. The absolute maximum is $2$ , attained at $x=-1$ and at $x=2$ .

Solution: The candidates are $0$ , $4$ , $9$ , and $12$ , with volumes $0$ , $18$ , $11$ , and $20$ . The maximum volume is $20$ at $t=12$ , which is larger than the local peak of $18$ at $t=4$ . The comparison is between accumulated volumes, not between rates, and the endpoint wins.

Common Traps & Failure Signals:

The rate compared (Signal: the largest rate offered as the maximum volume): the wrong function was tabulated.
The blank endpoint (Signal: a comparison of interior candidates only): an endpoint row was never filled.
The single location (Signal: one input reported for a tied extreme): a tie was reported as if it were unique.

6D Cognitive Error Diagnosis (Where Your Thinking Broke):

Column Empty: You compared before you evaluated.
Function Swapped: You tabulated the integrand instead of the integral, which is the accumulation version of answering a different question with correct arithmetic.
Tie Collapsed: You found the extreme value and stopped at the first input that produced it, which loses the second location on a part whose guideline does score both locations separately.

Past 10-Year AP Exam Appearance

Testing Frequency: Nearly Every Year; Seven released BC sets make the evaluation step the substance of the part, and in every one of them the function is supplied as a table of rates or as the graph of its own derivative rather than as a formula
10-Year Total Count: 7 of 9 Exams · 9 FRQ Parts; Counted where the scored work is converting supplied data into values at the candidates and comparing them
FRQ Years Appeared: 2017 (Q3c), 2018 (Q1d), 2019 (Q1c, Q3c), 2022 (Q1d), 2023 (Q4d), 2025 (Q1D, Q4D), 2026 (Q4D)

Practice Examples

MCQ 1

Four Values

A differentiable function has candidates $x=1$ , $x=3$ , $x=6$ , and $x=8$ on its interval, with values $f(1)=5$ , $f(3)=12$ , $f(6)=-2$ , and $f(8)=12$ . What is the absolute maximum?

MCQ 2

An Accumulation

A function is defined by $V(x)=\int_{0}^{x} r(t)\,dt$ with $V(0)=0$ , $V(2)=9$ , $V(5)=4$ , and $V(7)=6$ . On $[0,7]$ , where does $V$ attain its maximum?

MCQ 3

A Rate Trap

The rate $r(t)$ in gallons per minute is largest at $t=5$ . The accumulated volume is $V(x)=\int_{0}^{x} r(t)\,dt$ . Which statement must be true?

FRQ 1 • From Rates to Totals

Water flows into a tank at a rate $r(t)$ given in the table, where $t$ is measured in minutes and $r(t)$ in liters per minute. Let $W(x)=\int_{0}^{x} r(t)\,dt$ with $W(0)=0$ .

$t$	$0$	$3$	$6$	$9$	$12$
$r(t)$	$2$	$8$	$5$	$0$	$6$

(a) Use a left Riemann sum with the four subintervals to approximate $W(12)$ .

(b) It is known that $W(3)=15$ , $W(6)=34$ , and $W(9)=40$ . Explain why $W$ has a critical number at $t=9$ .

(c) Find the maximum value of $W$ on $[0,12]$ , using $W(9)=40$ and the approximation of $W(12)$ from part (a).

(d) A student compares the values of $r(t)$ and concludes that $W$ is largest at $t=3$ . Explain the error.

Cheatsheet

Evaluating Candidates

	What to compute	Common error
explicit formula	$f$ at each candidate	comparing $f'$ values
accumulation	running totals	comparing rates
table only	the listed values	interpolating
tie	both locations	one location
report	value plus location plus unit	value alone

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