Unit 5First Derivative Test for Increasing/Decreasing

Concept 4.245Built for May 2027 AP Exam

First Derivative Test for Increasing/Decreasing

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

The sign of the derivative decides the direction of the function. Where $f'(x)>0$ on an interval, $f$ is increasing there, and where $f'(x)<0$ , $f$ is decreasing. The proof is the Mean Value Theorem from 5.1, which turns a statement about signs into a statement about an average rate and then into an inequality between function values. A sign chart is the whole method: find the critical numbers, test one point in each interval, record the sign, and read the intervals off the chart. Two presentation habits carry most of the points. Write intervals with open endpoints, and justify with the sign of $f'$ rather than with the appearance of a graph.

On the AP Exam

What you’ll see: "On what intervals is $f$ increasing" or "find the open intervals on which the graph of $f$ is both increasing and concave up", usually attached to a given derivative or a given graph of $f'$ .
What you’ll do: Locate the critical numbers, build the sign chart from the factored derivative, then state the intervals with the sign that produces the requested direction.
Memory hook: Sign sets direction. Positive derivative, climbing function; negative derivative, falling function.
Use it on the exam: For MCQs: The distractors differ in the brackets, offering closed intervals where the derivative vanishes at an endpoint, and one open choice is usually the correct one. For FRQs: On 2018 Q3(c) the question asks for intervals where the graph is both increasing and concave up, so both signs must hold on the same interval, and each sign has to be justified separately.

Concept Review

The Sign Chart, and What It Encodes

Figure 5 — A sign chart under a number line. The derivative changes sign at each critical number, and the switch from plus to minus marks a local maximum while the switch from minus to plus marks a local minimum.

The chart has three rows of information: the critical numbers as boundaries, the sign of $f'$ between them, and the resulting direction of $f$ . Reading it left to right reproduces the shape of the graph without any plotting. The arrows in the figure are the direction of $f$ , not of $f'$ , and keeping those two straight is the first skill this topic asks for.

A sign chart under a number line with two critical numbers, showing plus, minus, plus regions with rising and falling arrows and the resulting local maximum and minimum

Why a Positive Derivative Means Increasing

Take $x_1<x_2$ in an interval where $f'>0$ . The Mean Value Theorem gives a $c$ between them with

$f'(c)=\frac{f(x_2)-f(x_1)}{x_2-x_1}$

The left side is positive and the denominator is positive, so the numerator is positive and $f(x_2)>f(x_1)$ . Since the two points were arbitrary, $f$ is increasing on the whole interval. The same computation with the sign reversed gives decreasing, and with $f'=0$ throughout it gives constancy. Every justification of an increasing interval is this theorem in disguise.

Building the Chart Without Errors

The chart is built from the factored derivative, and the factoring is what makes it reliable:

Step	What you produce	What it protects
factor $f'$ completely	a product of linear and quadratic factors	the exact list of critical numbers
mark the critical numbers	boundaries between test intervals	no missed region
test one point per interval	the sign, not the value	the direction on that whole interval
record the conclusion	interval plus direction	the sentence the rubric wants

A quadratic factor that never changes sign contributes no boundary, and a repeated root contributes no switch. Both facts come from the factored form, which is why an unfactored derivative is a liability rather than a shortcut.

Open Intervals, and Why the Brackets Matter

The conclusion is stated on open intervals, because at a critical number the derivative is zero and the function is momentarily neither rising nor falling in the sense the theorem guarantees. That is why the answer to an increasing/decreasing question is written $(a,b)$ and not $[a,b]$ . Some textbooks allow closed intervals by a separate argument, but the AP scoring consistently reads open intervals, and writing the closed version invites a losing argument with the reader.

Reading Direction From a Graph of $f'$

When the question supplies the graph of $f'$ rather than a formula, the sign chart is read directly: $f$ increases where the graph of $f'$ is above the axis, decreases where it is below, and has a local extreme where the graph crosses the axis. A touch without a crossing produces no extreme, which is the graph version of a repeated root. The full translation table lives in 5.17, and this paragraph is the part of it that belongs here.

Common Misconception Check

Myth 1: "The function increases where it is positive." Truth: The direction comes from the sign of the derivative, not from the sign of the function. A function can be positive and falling.
Myth 2: "Critical numbers belong in the interval of increase." Truth: The intervals are open. A critical number is a boundary between intervals, not part of either one.
Myth 3: "A zero of $f'$ always changes the direction." Truth: $f(x)=x^{3}$ has $f'(0)=0$ with no change of sign, so the function keeps increasing through that point.

Topic-Specific Study Method

Factor, Test, Read

Borrowed idea: A chart is only as good as the factoring that feeds it. Factor the derivative completely before anything else. List the boundaries in increasing order. Test one convenient point in each region, and only the sign matters. The test value never needs to be pretty. Only its sign enters the argument, and a sign is harder to get wrong than a simplification. So which test points should you pick, and when is a chart unnecessary? Pick numbers that make each factor obviously positive or negative, and skip the chart when the derivative arrives already factored.

Derivative form	Boundaries	Fastest test
$(x-1)(x+4)$	$-4$ and $1$	$x=-5$ , $x=0$ , $x=2$
$x^{2}(x-3)$	$3$ only, since $0$ is a double root	$x=-1$ , $x=1$ , $x=4$
$\dfrac{x-2}{x+1}$	$2$ , and the domain gap at $-1$	one point in each of the three regions
$e^{x}(x-2)$	$2$ only, since $e^{x}>0$	$x=0$ , $x=3$

① Write the boundary list in increasing order before testing anything. The order is what prevents a skipped region.

② Record the sign of each factor, not only the sign of the product. When two factors flip at once, the product's sign is easy to misjudge, and the per-factor record catches it.

③ End the answer with the direction written in words next to each interval. A list of intervals with no directions is a chart, not a conclusion.

Training & Error Diagnosis

Core Archetype (Simplest Form): Find the open intervals on which $f(x)=x^{3}-3x^{2}$ is increasing.

Standard Execution:

Factor. $f'(x)=3x^{2}-6x=3x(x-2)$ , so the critical numbers are $0$ and $2$ .

Test. $f'(-1)>0$ , $f'(1)<0$ , $f'(3)>0$ .

Read. The sign is positive on $(-\infty,0)$ and on $(2,\infty)$ .

State. $f$ is increasing on $(-\infty,0)$ and on $(2,\infty)$ , and decreasing on $(0,2)$ .

Exam Variation (Context / Parameter Shift): The graph of $f'$ , a continuous function, crosses the $x$ -axis at $x=-3$ from above to below and at $x=1$ from below to above, and it touches the axis at $x=4$ without crossing. On what intervals is $f$ increasing?

Solution: $f$ increases where $f'$ is above the axis. That happens on $(-\infty,-3)$ , where $f'$ starts positive, and on $(1,4)$ , where $f'$ is positive again after the crossing at $1$ . The touch at $x=4$ changes nothing, because $f'$ stays positive on both sides, so the interval continues to $4$ and then beyond it if the graph stays above the axis.

Common Traps & Failure Signals:

The sign mix-up (Signal: increasing intervals listed where $f$ itself is positive): the function's sign was read instead of its derivative's sign.
The closed bracket (Signal: $[0,2]$ offered as an interval of decrease): the open-interval convention was not applied.
The touch counted (Signal: an extreme claimed where $f'$ only touches the axis): a repeated root was treated as a sign change.

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

Sign Confused: You tracked the function, not the derivative.
Factoring Skipped: You tested points against an unfactored derivative, so the boundary list came from solving a quadratic by formula and the repeated root disappeared into the arithmetic.
Bracket Habit: You wrote closed intervals out of habit from the extreme-value questions, which is a convention error rather than a calculation error and costs the communication point.

Past 10-Year AP Exam Appearance

Testing Frequency: Most Years; Four released BC sets ask for a direction on an interval, and two of those combine the question with a concavity condition so that both signs must hold at once
10-Year Total Count: 4 of 9 Exams · 5 FRQ Parts; Counted where the scored answer is a direction on an interval, not where a sign is merely mentioned in passing
FRQ Years Appeared: 2017 (Q2d, Q3b), 2018 (Q3c), 2022 (Q3c), 2026 (Q4C)

Practice Examples

MCQ 1

A Cubic

On which intervals is $f(x)=x^{3}-6x^{2}+9x$ increasing?

MCQ 2

A Repeated Root

The derivative of a differentiable function is $f'(x)=x^{2}(x-4)$ . On what interval is $f$ decreasing?

MCQ 3

From a Graph of the Derivative

The graph of $f'$ lies below the $x$ -axis on $(0,2)$ and above it everywhere else. Which statement is correct?

FRQ 1 • Chart, Then Justify

Let $f(x)=x^{4}-4x^{3}$ .

(a) Find $f'(x)$ and factor it completely.

(b) Find all critical numbers of $f$ .

(c) Determine the open intervals on which $f$ is increasing and those on which it is decreasing.

(d) Justify your conclusion in part (c) using the sign of $f'$ and the theorem that connects that sign to the direction of $f$ .

Cheatsheet

Increasing and Decreasing

	What to write	Common error
boundaries	the factored zeros of $f'$	missing a repeated root
test	one point per interval, sign only	using function values
increasing	$f'>0$ there	reading the sign of $f$
decreasing	$f'<0$ there	reversing the correspondence
intervals	open, with directions in words	closed brackets

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Start Topic Practice Next Topic: 5.6 First Derivative Test Procedure

Unit 5First Derivative Test for Increasing/Decreasing

Concept 4.245Built for May 2027 AP Exam

First Derivative Test for Increasing/Decreasing

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

On the AP Exam

What you’ll see: "On what intervals is $f$ increasing" or "find the open intervals on which the graph of $f$ is both increasing and concave up", usually attached to a given derivative or a given graph of $f'$ .
What you’ll do: Locate the critical numbers, build the sign chart from the factored derivative, then state the intervals with the sign that produces the requested direction.
Memory hook: Sign sets direction. Positive derivative, climbing function; negative derivative, falling function.
Use it on the exam: For MCQs: The distractors differ in the brackets, offering closed intervals where the derivative vanishes at an endpoint, and one open choice is usually the correct one. For FRQs: On 2018 Q3(c) the question asks for intervals where the graph is both increasing and concave up, so both signs must hold on the same interval, and each sign has to be justified separately.

Concept Review

The Sign Chart, and What It Encodes

A sign chart under a number line with two critical numbers, showing plus, minus, plus regions with rising and falling arrows and the resulting local maximum and minimum

Why a Positive Derivative Means Increasing

Take $x_1<x_2$ in an interval where $f'>0$ . The Mean Value Theorem gives a $c$ between them with

$f'(c)=\frac{f(x_2)-f(x_1)}{x_2-x_1}$

Building the Chart Without Errors

The chart is built from the factored derivative, and the factoring is what makes it reliable:

Step	What you produce	What it protects
factor $f'$ completely	a product of linear and quadratic factors	the exact list of critical numbers
mark the critical numbers	boundaries between test intervals	no missed region
test one point per interval	the sign, not the value	the direction on that whole interval
record the conclusion	interval plus direction	the sentence the rubric wants

Open Intervals, and Why the Brackets Matter

Reading Direction From a Graph of $f'$

Common Misconception Check

Myth 1: "The function increases where it is positive." Truth: The direction comes from the sign of the derivative, not from the sign of the function. A function can be positive and falling.
Myth 2: "Critical numbers belong in the interval of increase." Truth: The intervals are open. A critical number is a boundary between intervals, not part of either one.
Myth 3: "A zero of $f'$ always changes the direction." Truth: $f(x)=x^{3}$ has $f'(0)=0$ with no change of sign, so the function keeps increasing through that point.

Topic-Specific Study Method

Factor, Test, Read

Derivative form	Boundaries	Fastest test
$(x-1)(x+4)$	$-4$ and $1$	$x=-5$ , $x=0$ , $x=2$
$x^{2}(x-3)$	$3$ only, since $0$ is a double root	$x=-1$ , $x=1$ , $x=4$
$\dfrac{x-2}{x+1}$	$2$ , and the domain gap at $-1$	one point in each of the three regions
$e^{x}(x-2)$	$2$ only, since $e^{x}>0$	$x=0$ , $x=3$

① Write the boundary list in increasing order before testing anything. The order is what prevents a skipped region.

② Record the sign of each factor, not only the sign of the product. When two factors flip at once, the product's sign is easy to misjudge, and the per-factor record catches it.

③ End the answer with the direction written in words next to each interval. A list of intervals with no directions is a chart, not a conclusion.

Training & Error Diagnosis

Core Archetype (Simplest Form): Find the open intervals on which $f(x)=x^{3}-3x^{2}$ is increasing.

Standard Execution:

Factor. $f'(x)=3x^{2}-6x=3x(x-2)$ , so the critical numbers are $0$ and $2$ .

Test. $f'(-1)>0$ , $f'(1)<0$ , $f'(3)>0$ .

Read. The sign is positive on $(-\infty,0)$ and on $(2,\infty)$ .

State. $f$ is increasing on $(-\infty,0)$ and on $(2,\infty)$ , and decreasing on $(0,2)$ .

Solution: $f$ increases where $f'$ is above the axis. That happens on $(-\infty,-3)$ , where $f'$ starts positive, and on $(1,4)$ , where $f'$ is positive again after the crossing at $1$ . The touch at $x=4$ changes nothing, because $f'$ stays positive on both sides, so the interval continues to $4$ and then beyond it if the graph stays above the axis.

Common Traps & Failure Signals:

The sign mix-up (Signal: increasing intervals listed where $f$ itself is positive): the function's sign was read instead of its derivative's sign.
The closed bracket (Signal: $[0,2]$ offered as an interval of decrease): the open-interval convention was not applied.
The touch counted (Signal: an extreme claimed where $f'$ only touches the axis): a repeated root was treated as a sign change.

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

Sign Confused: You tracked the function, not the derivative.
Factoring Skipped: You tested points against an unfactored derivative, so the boundary list came from solving a quadratic by formula and the repeated root disappeared into the arithmetic.
Bracket Habit: You wrote closed intervals out of habit from the extreme-value questions, which is a convention error rather than a calculation error and costs the communication point.

Past 10-Year AP Exam Appearance

Testing Frequency: Most Years; Four released BC sets ask for a direction on an interval, and two of those combine the question with a concavity condition so that both signs must hold at once
10-Year Total Count: 4 of 9 Exams · 5 FRQ Parts; Counted where the scored answer is a direction on an interval, not where a sign is merely mentioned in passing
FRQ Years Appeared: 2017 (Q2d, Q3b), 2018 (Q3c), 2022 (Q3c), 2026 (Q4C)

Practice Examples

MCQ 1

A Cubic

On which intervals is $f(x)=x^{3}-6x^{2}+9x$ increasing?

MCQ 2

A Repeated Root

The derivative of a differentiable function is $f'(x)=x^{2}(x-4)$ . On what interval is $f$ decreasing?

MCQ 3

From a Graph of the Derivative

The graph of $f'$ lies below the $x$ -axis on $(0,2)$ and above it everywhere else. Which statement is correct?

FRQ 1 • Chart, Then Justify

Let $f(x)=x^{4}-4x^{3}$ .

(a) Find $f'(x)$ and factor it completely.

(b) Find all critical numbers of $f$ .

(c) Determine the open intervals on which $f$ is increasing and those on which it is decreasing.

(d) Justify your conclusion in part (c) using the sign of $f'$ and the theorem that connects that sign to the direction of $f$ .

Cheatsheet

Increasing and Decreasing

	What to write	Common error
boundaries	the factored zeros of $f'$	missing a repeated root
test	one point per interval, sign only	using function values
increasing	$f'>0$ there	reading the sign of $f$
decreasing	$f'<0$ there	reversing the correspondence
intervals	open, with directions in words	closed brackets

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Start Topic Practice Next Topic: 5.6 First Derivative Test Procedure

Unit 5First Derivative Test for Increasing/Decreasing

Concept 4.245Built for May 2027 AP Exam

First Derivative Test for Increasing/Decreasing

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

On the AP Exam

What you’ll see: "On what intervals is $f$ increasing" or "find the open intervals on which the graph of $f$ is both increasing and concave up", usually attached to a given derivative or a given graph of $f'$ .
What you’ll do: Locate the critical numbers, build the sign chart from the factored derivative, then state the intervals with the sign that produces the requested direction.
Memory hook: Sign sets direction. Positive derivative, climbing function; negative derivative, falling function.
Use it on the exam: For MCQs: The distractors differ in the brackets, offering closed intervals where the derivative vanishes at an endpoint, and one open choice is usually the correct one. For FRQs: On 2018 Q3(c) the question asks for intervals where the graph is both increasing and concave up, so both signs must hold on the same interval, and each sign has to be justified separately.

Concept Review

The Sign Chart, and What It Encodes

A sign chart under a number line with two critical numbers, showing plus, minus, plus regions with rising and falling arrows and the resulting local maximum and minimum

Why a Positive Derivative Means Increasing

Take $x_1<x_2$ in an interval where $f'>0$ . The Mean Value Theorem gives a $c$ between them with

$f'(c)=\frac{f(x_2)-f(x_1)}{x_2-x_1}$

Building the Chart Without Errors

The chart is built from the factored derivative, and the factoring is what makes it reliable:

Step	What you produce	What it protects
factor $f'$ completely	a product of linear and quadratic factors	the exact list of critical numbers
mark the critical numbers	boundaries between test intervals	no missed region
test one point per interval	the sign, not the value	the direction on that whole interval
record the conclusion	interval plus direction	the sentence the rubric wants

Open Intervals, and Why the Brackets Matter

Reading Direction From a Graph of $f'$

Common Misconception Check

Myth 1: "The function increases where it is positive." Truth: The direction comes from the sign of the derivative, not from the sign of the function. A function can be positive and falling.
Myth 2: "Critical numbers belong in the interval of increase." Truth: The intervals are open. A critical number is a boundary between intervals, not part of either one.
Myth 3: "A zero of $f'$ always changes the direction." Truth: $f(x)=x^{3}$ has $f'(0)=0$ with no change of sign, so the function keeps increasing through that point.

Topic-Specific Study Method

Factor, Test, Read

Derivative form	Boundaries	Fastest test
$(x-1)(x+4)$	$-4$ and $1$	$x=-5$ , $x=0$ , $x=2$
$x^{2}(x-3)$	$3$ only, since $0$ is a double root	$x=-1$ , $x=1$ , $x=4$
$\dfrac{x-2}{x+1}$	$2$ , and the domain gap at $-1$	one point in each of the three regions
$e^{x}(x-2)$	$2$ only, since $e^{x}>0$	$x=0$ , $x=3$

① Write the boundary list in increasing order before testing anything. The order is what prevents a skipped region.

② Record the sign of each factor, not only the sign of the product. When two factors flip at once, the product's sign is easy to misjudge, and the per-factor record catches it.

③ End the answer with the direction written in words next to each interval. A list of intervals with no directions is a chart, not a conclusion.

Training & Error Diagnosis

Core Archetype (Simplest Form): Find the open intervals on which $f(x)=x^{3}-3x^{2}$ is increasing.

Standard Execution:

Factor. $f'(x)=3x^{2}-6x=3x(x-2)$ , so the critical numbers are $0$ and $2$ .

Test. $f'(-1)>0$ , $f'(1)<0$ , $f'(3)>0$ .

Read. The sign is positive on $(-\infty,0)$ and on $(2,\infty)$ .

State. $f$ is increasing on $(-\infty,0)$ and on $(2,\infty)$ , and decreasing on $(0,2)$ .

Solution: $f$ increases where $f'$ is above the axis. That happens on $(-\infty,-3)$ , where $f'$ starts positive, and on $(1,4)$ , where $f'$ is positive again after the crossing at $1$ . The touch at $x=4$ changes nothing, because $f'$ stays positive on both sides, so the interval continues to $4$ and then beyond it if the graph stays above the axis.

Common Traps & Failure Signals:

The sign mix-up (Signal: increasing intervals listed where $f$ itself is positive): the function's sign was read instead of its derivative's sign.
The closed bracket (Signal: $[0,2]$ offered as an interval of decrease): the open-interval convention was not applied.
The touch counted (Signal: an extreme claimed where $f'$ only touches the axis): a repeated root was treated as a sign change.

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

Sign Confused: You tracked the function, not the derivative.
Factoring Skipped: You tested points against an unfactored derivative, so the boundary list came from solving a quadratic by formula and the repeated root disappeared into the arithmetic.
Bracket Habit: You wrote closed intervals out of habit from the extreme-value questions, which is a convention error rather than a calculation error and costs the communication point.

Past 10-Year AP Exam Appearance

Testing Frequency: Most Years; Four released BC sets ask for a direction on an interval, and two of those combine the question with a concavity condition so that both signs must hold at once
10-Year Total Count: 4 of 9 Exams · 5 FRQ Parts; Counted where the scored answer is a direction on an interval, not where a sign is merely mentioned in passing
FRQ Years Appeared: 2017 (Q2d, Q3b), 2018 (Q3c), 2022 (Q3c), 2026 (Q4C)

Practice Examples

MCQ 1

A Cubic

On which intervals is $f(x)=x^{3}-6x^{2}+9x$ increasing?

MCQ 2

A Repeated Root

The derivative of a differentiable function is $f'(x)=x^{2}(x-4)$ . On what interval is $f$ decreasing?

MCQ 3

From a Graph of the Derivative

The graph of $f'$ lies below the $x$ -axis on $(0,2)$ and above it everywhere else. Which statement is correct?

FRQ 1 • Chart, Then Justify

Let $f(x)=x^{4}-4x^{3}$ .

(a) Find $f'(x)$ and factor it completely.

(b) Find all critical numbers of $f$ .

(c) Determine the open intervals on which $f$ is increasing and those on which it is decreasing.

(d) Justify your conclusion in part (c) using the sign of $f'$ and the theorem that connects that sign to the direction of $f$ .

Cheatsheet

Increasing and Decreasing

	What to write	Common error
boundaries	the factored zeros of $f'$	missing a repeated root
test	one point per interval, sign only	using function values
increasing	$f'>0$ there	reading the sign of $f$
decreasing	$f'<0$ there	reversing the correspondence
intervals	open, with directions in words	closed brackets

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