Unit 5Finding Critical Numbers

Concept 4.271Built for May 2027 AP Exam

Finding Critical Numbers

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

A critical number of $f$ is an input $c$ in the domain of $f$ where either $f'(c)=0$ or $f'(c)$ does not exist. Both halves of that definition matter, and the domain clause is the part most responses drop. Critical numbers are the interior candidates for a local extreme, which is why the derivative tests and the candidates test both begin by finding them. Finding them is mechanical: differentiate, solve for zeros, then ask separately whether the derivative fails anywhere the function still lives. Endpoints are never critical numbers, even though they are candidates for absolute extremes, and the two lists are not the same list.

On the AP Exam

What you’ll see: A request for "the critical numbers" or "the value of $t$ at which $f$ has a critical point", sometimes inside a motion or polar question rather than a bare function.
What you’ll do: Differentiate, solve $f'(x)=0$ , then check the places where $f'$ is undefined while $f$ is defined, and report only those values that lie in the stated domain.
Memory hook: Zero or absent, never the edge. Both failures of the derivative qualify, and the endpoints never do.
Use it on the exam: For MCQs: The usual trap is a value where the function itself is undefined, such as a root of a denominator that also annihilates the numerator, and that value is not in the domain. For FRQs: On 2024 Q3(b) the question restricts attention to $0<t<5$ and asks for the time at which $H$ has a critical point, so the answer must come from the stated interval and not from all real solutions.

Concept Review

The Definition, in Three Clauses

Figure 4 — Three ways a point becomes critical. A flat tangent, a corner, and a vertical tangent all qualify, and each one sits inside the domain of the function.

A number $c$ is a critical number of $f$ when all three of these hold:

$c$ is in the domain of $f$ ,
$f$ is defined on an open interval around $c$ ,
either $f'(c)=0$ or $f'(c)$ does not exist.

Clause two is what makes the definition local. Clause one is what keeps vertical asymptotes off the list. A point where the function blows up is not a critical number, because there is no function value there to compare.

Three small panels showing a smooth turn with zero slope, a corner where the derivative fails, and a vertical tangent where the derivative grows without bound

The Procedure, and Where It Breaks

Differentiate, then solve $f'(x)=0$ , then inspect the domain of $f'$ against the domain of $f$ . The third step is where responses lose points, because the interesting failures are easy to overlook:

Function	$f'(x)$	Critical numbers
$x^{3}-3x$	$3x^{2}-3$	$x=\pm 1$
$\lvert x-2\rvert$	$\pm 1$ away from $2$	$x=2$ , where $f'$ fails
$x^{2/3}$	$\frac{2}{3}x^{-1/3}$	$x=0$ , where $f'$ is unbounded
$\frac{1}{x}$	$-\frac{1}{x^{2}}$	none, since $0$ is not in the domain

The last row is the one worth memorising. The derivative is never zero, and the one place it fails is a point the function never reaches, so the function has no critical numbers at all.

Why the Definition Includes the Absent Derivative

If the derivative fails at a point where the function is defined, something visible is happening there: a corner, a cusp, or a vertical tangent. All three can host a local extreme, and the corner case is exactly how $\lvert x\rvert$ gets its minimum. A definition that only counted solutions of $f'(x)=0$ would miss that, so the definition counts both.

Critical Numbers Versus Candidates

The two lists answer different questions and are built differently:

List	Contains	Used for
critical numbers	interior points where $f'$ is zero or absent	local extremes, sign charts, the tests in 5.6 and 5.11
candidates for absolute extremes	critical numbers plus both endpoints	the closed-interval search of 5.7

An endpoint can be the location of an absolute maximum and still never be a critical number, because the derivative there is one-sided. Keeping the lists separate prevents the most common vocabulary error in this unit.

Reading a Graph for Critical Numbers

On a graph of $f$ , the critical numbers are the $x$ -values where the tangent is horizontal or where the curve has a corner or a vertical tangent. On a graph of $f'$ , they are the $x$ -values where $f'$ crosses or touches the axis, together with the $x$ -values where $f'$ itself has a discontinuity while $f$ stays defined. The second reading is the one that appears in graph-translation questions, and it is the reason 5.17 exists.

Common Misconception Check

Myth 1: "Critical numbers are the solutions of $f'(x)=0$ ." Truth: The definition also includes points where $f'$ fails to exist, and $\lvert x\rvert$ at $0$ is the standard example.
Myth 2: "Endpoints are critical numbers." Truth: They are candidates for absolute extremes, but the definition requires an open interval around the point, which an endpoint does not have.
Myth 3: "Every critical number is a local extreme." Truth: $f(x)=x^{3}$ has a critical number at $0$ and no extreme there. The classification work belongs to 5.6 and 5.11.

Topic-Specific Study Method

Differentiate, Solve, Then Check the Domain

Borrowed idea: Check domains. Write the domain of $f$ and the domain of $f'$ side by side. The critical numbers live where both agree, minus the zeros of the derivative. That agreement is what separates a complete answer from a partial one, because the failures of the derivative are invisible in the algebra of $f'(x)=0$ . So which functions should make you suspicious before you even start, and what should you look for in them?

Signal in the function	What to expect	What to write
polynomial	solutions of $f'=0$ only	the factored derivative, then the roots
absolute value	a corner inside	the corner point, plus the pieces where $f'=\pm 1$
fractional power such as $x^{2/3}$	a vertical tangent at the base point	the point, with the note that $f'$ is unbounded
rational function	a possible domain gap	check whether the gap is in the domain of $f$

① Factor the derivative completely before solving. An unfactored quadratic hides the count, and the count is usually what the question is really testing.

② Say which of the two clauses each answer satisfies. "Since $f'(0)$ does not exist" and "since $f'(3)=0$ " are different justifications, and the reader is checking that you know the difference.

③ Filter by the domain the question states. A critical number outside the given interval is not an answer, even when it satisfies both clauses.

Training & Error Diagnosis

Core Archetype (Simplest Form): Find the critical numbers of $f(x)=x^{3}-6x^{2}+9x$ .

Standard Execution:

Differentiate. $f'(x)=3x^{2}-12x+9=3(x-1)(x-3)$ .

Solve. $f'(x)=0$ gives $x=1$ and $x=3$ .

Check. $f$ is a polynomial, so its domain is all reals and $f'$ exists everywhere.

Report. The critical numbers are $x=1$ and $x=3$ .

Exam Variation (Context / Parameter Shift): A function is defined by $g(x)=x^{2/3}(x-5)$ on $[-1,8]$ . Find all critical numbers of $g$ .

Solution: Rewrite $g(x)=x^{5/3}-5x^{2/3}$ , so $g'(x)=\frac{5}{3}x^{2/3}-\frac{10}{3}x^{-1/3}=\frac{5(x-2)}{3x^{1/3}}$ . Setting the numerator to zero gives $x=2$ . The derivative fails at $x=0$ , which is in the domain of $g$ , so $x=0$ is also critical. Both lie in $[-1,8]$ , and both are answers.

Common Traps & Failure Signals:

The halved definition (Signal: only the zeros of $f'$ reported): the absent-derivative clause was never applied, so corners and cusps were lost.
The domain gap (Signal: a critical number where $f$ is undefined): a zero of the denominator was promoted to a critical number without checking the domain of $f$ .
The endpoint promoted (Signal: $x=a$ listed for an interval $[a,b]$ ): the candidate list for absolute extremes was confused with the definition of a critical number.

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

Half the Definition: You solved the zeros and stopped there.
Domain Unchecked: You accepted a solution of the derivative equation as a critical number without asking whether the function is defined there, which is how an asymptote gets mistaken for a critical point.
Vocabulary Slip: You called an endpoint a critical number, which is a definitional error rather than a computational one and is easy for a reader to spot.

Past 10-Year AP Exam Appearance

Testing Frequency: Rare in FRQs; Three released BC sets ask for a critical point by name, and two of those do it inside a differential-equation or polar question rather than with a bare function rule
10-Year Total Count: 3 of 9 Exams · 4 FRQ Parts; Counted where the scored answer is the location of a critical point, not where a sign chart is being built
FRQ Years Appeared: 2017 (Q5b), 2024 (Q3b, Q4b), 2026 (Q2C)

Practice Examples

MCQ 1

A Polynomial

How many critical numbers does $f(x)=x^{4}-4x^{3}$ have?

MCQ 2

An Absolute Value

What are the critical numbers of $f(x)=\lvert x+3\rvert$ ?

MCQ 3

A Domain Trap

How many critical numbers does $f(x)=\dfrac{1}{x-2}$ have?

FRQ 1 • Two Clauses at Work

Let $f(x)=x^{2/3}(x-6)$ on the interval $[-1,10]$ .

(a) Find $f'(x)$ and write it as a single fraction.

(b) Find all critical numbers of $f$ in $(-1,10)$ .

(c) Explain why $x=0$ qualifies under one clause of the definition rather than the other.

(d) A student lists $x=-1$ and $x=10$ as critical numbers. Explain why both are wrong.

Cheatsheet

Critical Numbers

	Test	Common error
in the domain of $f$	substitute and check	promoting a vertical asymptote
interior point	an open interval around it lies in the domain	listing endpoints
$f'(c)=0$	factor and solve	missing a repeated root
$f'(c)$ absent	corners, cusps, vertical tangents	never checking
not an extreme	$x^{3}$ at $0$	assuming every critical number is a peak or valley

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Unit 5Finding Critical Numbers

Concept 4.271Built for May 2027 AP Exam

Finding Critical Numbers

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

On the AP Exam

What you’ll see: A request for "the critical numbers" or "the value of $t$ at which $f$ has a critical point", sometimes inside a motion or polar question rather than a bare function.
What you’ll do: Differentiate, solve $f'(x)=0$ , then check the places where $f'$ is undefined while $f$ is defined, and report only those values that lie in the stated domain.
Memory hook: Zero or absent, never the edge. Both failures of the derivative qualify, and the endpoints never do.
Use it on the exam: For MCQs: The usual trap is a value where the function itself is undefined, such as a root of a denominator that also annihilates the numerator, and that value is not in the domain. For FRQs: On 2024 Q3(b) the question restricts attention to $0<t<5$ and asks for the time at which $H$ has a critical point, so the answer must come from the stated interval and not from all real solutions.

Concept Review

The Definition, in Three Clauses

Figure 4 — Three ways a point becomes critical. A flat tangent, a corner, and a vertical tangent all qualify, and each one sits inside the domain of the function.

A number $c$ is a critical number of $f$ when all three of these hold:

$c$ is in the domain of $f$ ,
$f$ is defined on an open interval around $c$ ,
either $f'(c)=0$ or $f'(c)$ does not exist.

Three small panels showing a smooth turn with zero slope, a corner where the derivative fails, and a vertical tangent where the derivative grows without bound

The Procedure, and Where It Breaks

Differentiate, then solve $f'(x)=0$ , then inspect the domain of $f'$ against the domain of $f$ . The third step is where responses lose points, because the interesting failures are easy to overlook:

Function	$f'(x)$	Critical numbers
$x^{3}-3x$	$3x^{2}-3$	$x=\pm 1$
$\lvert x-2\rvert$	$\pm 1$ away from $2$	$x=2$ , where $f'$ fails
$x^{2/3}$	$\frac{2}{3}x^{-1/3}$	$x=0$ , where $f'$ is unbounded
$\frac{1}{x}$	$-\frac{1}{x^{2}}$	none, since $0$ is not in the domain

The last row is the one worth memorising. The derivative is never zero, and the one place it fails is a point the function never reaches, so the function has no critical numbers at all.

Why the Definition Includes the Absent Derivative

Critical Numbers Versus Candidates

The two lists answer different questions and are built differently:

List	Contains	Used for
critical numbers	interior points where $f'$ is zero or absent	local extremes, sign charts, the tests in 5.6 and 5.11
candidates for absolute extremes	critical numbers plus both endpoints	the closed-interval search of 5.7

Reading a Graph for Critical Numbers

Common Misconception Check

Myth 1: "Critical numbers are the solutions of $f'(x)=0$ ." Truth: The definition also includes points where $f'$ fails to exist, and $\lvert x\rvert$ at $0$ is the standard example.
Myth 2: "Endpoints are critical numbers." Truth: They are candidates for absolute extremes, but the definition requires an open interval around the point, which an endpoint does not have.
Myth 3: "Every critical number is a local extreme." Truth: $f(x)=x^{3}$ has a critical number at $0$ and no extreme there. The classification work belongs to 5.6 and 5.11.

Topic-Specific Study Method

Differentiate, Solve, Then Check the Domain

Signal in the function	What to expect	What to write
polynomial	solutions of $f'=0$ only	the factored derivative, then the roots
absolute value	a corner inside	the corner point, plus the pieces where $f'=\pm 1$
fractional power such as $x^{2/3}$	a vertical tangent at the base point	the point, with the note that $f'$ is unbounded
rational function	a possible domain gap	check whether the gap is in the domain of $f$

① Factor the derivative completely before solving. An unfactored quadratic hides the count, and the count is usually what the question is really testing.

② Say which of the two clauses each answer satisfies. "Since $f'(0)$ does not exist" and "since $f'(3)=0$ " are different justifications, and the reader is checking that you know the difference.

③ Filter by the domain the question states. A critical number outside the given interval is not an answer, even when it satisfies both clauses.

Training & Error Diagnosis

Core Archetype (Simplest Form): Find the critical numbers of $f(x)=x^{3}-6x^{2}+9x$ .

Standard Execution:

Differentiate. $f'(x)=3x^{2}-12x+9=3(x-1)(x-3)$ .

Solve. $f'(x)=0$ gives $x=1$ and $x=3$ .

Check. $f$ is a polynomial, so its domain is all reals and $f'$ exists everywhere.

Report. The critical numbers are $x=1$ and $x=3$ .

Exam Variation (Context / Parameter Shift): A function is defined by $g(x)=x^{2/3}(x-5)$ on $[-1,8]$ . Find all critical numbers of $g$ .

Solution: Rewrite $g(x)=x^{5/3}-5x^{2/3}$ , so $g'(x)=\frac{5}{3}x^{2/3}-\frac{10}{3}x^{-1/3}=\frac{5(x-2)}{3x^{1/3}}$ . Setting the numerator to zero gives $x=2$ . The derivative fails at $x=0$ , which is in the domain of $g$ , so $x=0$ is also critical. Both lie in $[-1,8]$ , and both are answers.

Common Traps & Failure Signals:

The halved definition (Signal: only the zeros of $f'$ reported): the absent-derivative clause was never applied, so corners and cusps were lost.
The domain gap (Signal: a critical number where $f$ is undefined): a zero of the denominator was promoted to a critical number without checking the domain of $f$ .
The endpoint promoted (Signal: $x=a$ listed for an interval $[a,b]$ ): the candidate list for absolute extremes was confused with the definition of a critical number.

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

Half the Definition: You solved the zeros and stopped there.
Domain Unchecked: You accepted a solution of the derivative equation as a critical number without asking whether the function is defined there, which is how an asymptote gets mistaken for a critical point.
Vocabulary Slip: You called an endpoint a critical number, which is a definitional error rather than a computational one and is easy for a reader to spot.

Past 10-Year AP Exam Appearance

Testing Frequency: Rare in FRQs; Three released BC sets ask for a critical point by name, and two of those do it inside a differential-equation or polar question rather than with a bare function rule
10-Year Total Count: 3 of 9 Exams · 4 FRQ Parts; Counted where the scored answer is the location of a critical point, not where a sign chart is being built
FRQ Years Appeared: 2017 (Q5b), 2024 (Q3b, Q4b), 2026 (Q2C)

Practice Examples

MCQ 1

A Polynomial

How many critical numbers does $f(x)=x^{4}-4x^{3}$ have?

MCQ 2

An Absolute Value

What are the critical numbers of $f(x)=\lvert x+3\rvert$ ?

MCQ 3

A Domain Trap

How many critical numbers does $f(x)=\dfrac{1}{x-2}$ have?

FRQ 1 • Two Clauses at Work

Let $f(x)=x^{2/3}(x-6)$ on the interval $[-1,10]$ .

(a) Find $f'(x)$ and write it as a single fraction.

(b) Find all critical numbers of $f$ in $(-1,10)$ .

(c) Explain why $x=0$ qualifies under one clause of the definition rather than the other.

(d) A student lists $x=-1$ and $x=10$ as critical numbers. Explain why both are wrong.

Cheatsheet

Critical Numbers

	Test	Common error
in the domain of $f$	substitute and check	promoting a vertical asymptote
interior point	an open interval around it lies in the domain	listing endpoints
$f'(c)=0$	factor and solve	missing a repeated root
$f'(c)$ absent	corners, cusps, vertical tangents	never checking
not an extreme	$x^{3}$ at $0$	assuming every critical number is a peak or valley

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Unit 5Finding Critical Numbers

Concept 4.271Built for May 2027 AP Exam

Finding Critical Numbers

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

On the AP Exam

What you’ll see: A request for "the critical numbers" or "the value of $t$ at which $f$ has a critical point", sometimes inside a motion or polar question rather than a bare function.
What you’ll do: Differentiate, solve $f'(x)=0$ , then check the places where $f'$ is undefined while $f$ is defined, and report only those values that lie in the stated domain.
Memory hook: Zero or absent, never the edge. Both failures of the derivative qualify, and the endpoints never do.
Use it on the exam: For MCQs: The usual trap is a value where the function itself is undefined, such as a root of a denominator that also annihilates the numerator, and that value is not in the domain. For FRQs: On 2024 Q3(b) the question restricts attention to $0<t<5$ and asks for the time at which $H$ has a critical point, so the answer must come from the stated interval and not from all real solutions.

Concept Review

The Definition, in Three Clauses

Figure 4 — Three ways a point becomes critical. A flat tangent, a corner, and a vertical tangent all qualify, and each one sits inside the domain of the function.

A number $c$ is a critical number of $f$ when all three of these hold:

$c$ is in the domain of $f$ ,
$f$ is defined on an open interval around $c$ ,
either $f'(c)=0$ or $f'(c)$ does not exist.

Three small panels showing a smooth turn with zero slope, a corner where the derivative fails, and a vertical tangent where the derivative grows without bound

The Procedure, and Where It Breaks

Differentiate, then solve $f'(x)=0$ , then inspect the domain of $f'$ against the domain of $f$ . The third step is where responses lose points, because the interesting failures are easy to overlook:

Function	$f'(x)$	Critical numbers
$x^{3}-3x$	$3x^{2}-3$	$x=\pm 1$
$\lvert x-2\rvert$	$\pm 1$ away from $2$	$x=2$ , where $f'$ fails
$x^{2/3}$	$\frac{2}{3}x^{-1/3}$	$x=0$ , where $f'$ is unbounded
$\frac{1}{x}$	$-\frac{1}{x^{2}}$	none, since $0$ is not in the domain

The last row is the one worth memorising. The derivative is never zero, and the one place it fails is a point the function never reaches, so the function has no critical numbers at all.

Why the Definition Includes the Absent Derivative

Critical Numbers Versus Candidates

The two lists answer different questions and are built differently:

List	Contains	Used for
critical numbers	interior points where $f'$ is zero or absent	local extremes, sign charts, the tests in 5.6 and 5.11
candidates for absolute extremes	critical numbers plus both endpoints	the closed-interval search of 5.7

Reading a Graph for Critical Numbers

Common Misconception Check

Myth 1: "Critical numbers are the solutions of $f'(x)=0$ ." Truth: The definition also includes points where $f'$ fails to exist, and $\lvert x\rvert$ at $0$ is the standard example.
Myth 2: "Endpoints are critical numbers." Truth: They are candidates for absolute extremes, but the definition requires an open interval around the point, which an endpoint does not have.
Myth 3: "Every critical number is a local extreme." Truth: $f(x)=x^{3}$ has a critical number at $0$ and no extreme there. The classification work belongs to 5.6 and 5.11.

Topic-Specific Study Method

Differentiate, Solve, Then Check the Domain

Signal in the function	What to expect	What to write
polynomial	solutions of $f'=0$ only	the factored derivative, then the roots
absolute value	a corner inside	the corner point, plus the pieces where $f'=\pm 1$
fractional power such as $x^{2/3}$	a vertical tangent at the base point	the point, with the note that $f'$ is unbounded
rational function	a possible domain gap	check whether the gap is in the domain of $f$

① Factor the derivative completely before solving. An unfactored quadratic hides the count, and the count is usually what the question is really testing.

② Say which of the two clauses each answer satisfies. "Since $f'(0)$ does not exist" and "since $f'(3)=0$ " are different justifications, and the reader is checking that you know the difference.

③ Filter by the domain the question states. A critical number outside the given interval is not an answer, even when it satisfies both clauses.

Training & Error Diagnosis

Core Archetype (Simplest Form): Find the critical numbers of $f(x)=x^{3}-6x^{2}+9x$ .

Standard Execution:

Differentiate. $f'(x)=3x^{2}-12x+9=3(x-1)(x-3)$ .

Solve. $f'(x)=0$ gives $x=1$ and $x=3$ .

Check. $f$ is a polynomial, so its domain is all reals and $f'$ exists everywhere.

Report. The critical numbers are $x=1$ and $x=3$ .

Exam Variation (Context / Parameter Shift): A function is defined by $g(x)=x^{2/3}(x-5)$ on $[-1,8]$ . Find all critical numbers of $g$ .

Solution: Rewrite $g(x)=x^{5/3}-5x^{2/3}$ , so $g'(x)=\frac{5}{3}x^{2/3}-\frac{10}{3}x^{-1/3}=\frac{5(x-2)}{3x^{1/3}}$ . Setting the numerator to zero gives $x=2$ . The derivative fails at $x=0$ , which is in the domain of $g$ , so $x=0$ is also critical. Both lie in $[-1,8]$ , and both are answers.

Common Traps & Failure Signals:

The halved definition (Signal: only the zeros of $f'$ reported): the absent-derivative clause was never applied, so corners and cusps were lost.
The domain gap (Signal: a critical number where $f$ is undefined): a zero of the denominator was promoted to a critical number without checking the domain of $f$ .
The endpoint promoted (Signal: $x=a$ listed for an interval $[a,b]$ ): the candidate list for absolute extremes was confused with the definition of a critical number.

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

Half the Definition: You solved the zeros and stopped there.
Domain Unchecked: You accepted a solution of the derivative equation as a critical number without asking whether the function is defined there, which is how an asymptote gets mistaken for a critical point.
Vocabulary Slip: You called an endpoint a critical number, which is a definitional error rather than a computational one and is easy for a reader to spot.

Past 10-Year AP Exam Appearance

Testing Frequency: Rare in FRQs; Three released BC sets ask for a critical point by name, and two of those do it inside a differential-equation or polar question rather than with a bare function rule
10-Year Total Count: 3 of 9 Exams · 4 FRQ Parts; Counted where the scored answer is the location of a critical point, not where a sign chart is being built
FRQ Years Appeared: 2017 (Q5b), 2024 (Q3b, Q4b), 2026 (Q2C)

Practice Examples

MCQ 1

A Polynomial

How many critical numbers does $f(x)=x^{4}-4x^{3}$ have?

MCQ 2

An Absolute Value

What are the critical numbers of $f(x)=\lvert x+3\rvert$ ?

MCQ 3

A Domain Trap

How many critical numbers does $f(x)=\dfrac{1}{x-2}$ have?

FRQ 1 • Two Clauses at Work

Let $f(x)=x^{2/3}(x-6)$ on the interval $[-1,10]$ .

(a) Find $f'(x)$ and write it as a single fraction.

(b) Find all critical numbers of $f$ in $(-1,10)$ .

(c) Explain why $x=0$ qualifies under one clause of the definition rather than the other.

(d) A student lists $x=-1$ and $x=10$ as critical numbers. Explain why both are wrong.

Cheatsheet

Critical Numbers

	Test	Common error
in the domain of $f$	substitute and check	promoting a vertical asymptote
interior point	an open interval around it lies in the domain	listing endpoints
$f'(c)=0$	factor and solve	missing a repeated root
$f'(c)$ absent	corners, cusps, vertical tangents	never checking
not an extreme	$x^{3}$ at $0$	assuming every critical number is a peak or valley

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