Unit 5First Derivative Test Procedure

Concept 4.213Built for May 2027 AP Exam

First Derivative Test Procedure

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

The first derivative test classifies a critical number by watching the sign of the derivative on either side of it. If $f'$ changes from positive to negative at $c$ , the function rises into $c$ and falls away from it, so $c$ is a local maximum. If the sign changes from negative to positive, $c$ is a local minimum. If there is no change of sign, the point is neither. The procedure is short, and the scoring lives in the report: name the sign change, name the point, and name the type of extreme. A response that skips the sign-change sentence has found a number and abandoned the justification.

On the AP Exam

What you’ll see: "Does $f$ have a relative minimum, a relative maximum, or neither at $x=c$ ? Give a reason" — or a request to classify every critical point found in an earlier part.
What you’ll do: Build or read the sign chart around $c$ , then write the three-part sentence that names the change, the location, and the classification.
Memory hook: Plus to minus, peak. Minus to plus, pit. The order of the signs names the type.
Use it on the exam: For MCQs: The distractors often classify by the sign of $f''$ at the point when $f''$ is unavailable or zero, so the derivative test is the one that always works. For FRQs: On 2023 Q4(a) the guideline awards the point for the sign change of $f'$ across $x=6$ , and a response that only reports "minimum" without the sign evidence earns nothing.

Concept Review

The Three Outcomes

Figure 6 — The same flat tangent, three different neighbourhoods. What separates them is not the value of $f'$ at the point, which is zero in all three cases, but the sign of $f'$ on either side.

Behaviour of $f'$ at $c$	Shape of $f$ near $c$	Classification
$+$ then $-$	rises, then falls	local maximum
$-$ then $+$	falls, then rises	local minimum
$+$ then $+$	keeps climbing	neither
$-$ then $-$	keeps falling	neither

Note what the table does not use: the value $f'(c)$ itself. Every row has $f'(c)=0$ or $f'(c)$ undefined, so the value carries no information about the type. Only the sign pattern does.

Three panels: a curve rising then falling with a horizontal tangent at the peak, a curve falling then rising with a horizontal tangent at the valley, and a monotone curve with a horizontal tangent that changes nothing

The Sentence That Earns the Point

The justification has three slots, and all three are scored as one idea:

Because $f'(x)>0$ for $x<6$ and $f'(x)<0$ for $x>6$ , $f'$ changes from positive to negative at $x=6$ , so $f$ has a local maximum at $x=6$ .

Compare it with the common short version, " $f$ has a maximum at $x=6$ ". The short version reports a conclusion and hides the evidence. The long version shows the reader the sign pattern, and the sign pattern is what the rubric is checking for.

When the Derivative Does Not Exist

The test also applies at a critical number where $f'$ fails, provided the sign of $f'$ is known on both sides. For $f(x)=\lvert x\rvert$ the derivative is $-1$ on the left and $+1$ on the right, so the corner at $0$ is a local minimum by the same reasoning. The test never requires $f'(c)$ to exist; it requires the signs nearby to be readable.

Local Versus Absolute

The first derivative test classifies local extremes, also called relative extremes. It says nothing about absolute extremes, because a local maximum can be beaten by a value elsewhere on the interval. That comparison is the candidates test of 5.7. Keeping the two words separate prevents the most expensive vocabulary error in this unit, since a local maximum reported as an absolute maximum loses the justification point even when the location is right.

Reading the Test off a Graph of $f'$

If the graph of $f'$ is given, the classification is a crossing question. A crossing from above the axis to below it is a local maximum of $f$ , and a crossing from below to above is a local minimum. A touch that does not cross is neither. This is why graph-translation questions are really sign-chart questions in disguise, and it is the same reading that 5.17 formalises.

Common Misconception Check

Myth 1: "A zero of $f'$ is a maximum or a minimum." Truth: It is a candidate. The test decides, and $f(x)=x^{3}$ at $0$ shows that the answer can be neither.
Myth 2: "The second derivative decides everything the first derivative test decides." Truth: When $f''(c)=0$ the second derivative test is inconclusive, and the first derivative test still works. That is why 5.11 sends the inconclusive cases back here.
Myth 3: "The sign of $f'(c)$ tells you the type." Truth: At the critical number the derivative is zero or absent. The type comes from the signs on the two sides.

Topic-Specific Study Method

Name the Change, Then the Type

Borrowed idea: Write the sign change before you write the classification. The order forces you to look at both sides, which is the only part of the method that carries information. Then write the location. Then write the type. Three short clauses, and the sentence is complete. So what does a complete answer look like when the derivative is given only as a graph? The same three clauses, with "above the axis" and "below the axis" replacing the plus and minus signs.

Given	Evidence to quote	Sentence shape
factored $f'$	signs of the factors on each side	" $f'$ changes from $+$ to $-$ at $x=c$ "
graph of $f'$	crossing direction at the axis	" $f'$ crosses from above to below at $x=c$ "
corner in $f$	one-sided slopes	" $f'$ changes from negative to positive at $x=c$ "
$f''(c)=0$	sign chart of $f'$	"the second derivative test is inconclusive, so..."

① Test one point on each side of $c$ , not at $c$ . A test value at the critical number returns zero and tells you nothing.

② Say "local" or "relative" in the answer. The unqualified word "maximum" reads as absolute to a grader, and that is a different claim.

③ If a critical number produces neither, say so explicitly. "Neither" is a scored answer, and silence reads as an unfinished part.

Training & Error Diagnosis

Core Archetype (Simplest Form): Classify the critical numbers of $f(x)=x^{3}-3x$ .

Standard Execution:

Candidates. $f'(x)=3(x-1)(x+1)$ , so the critical numbers are $\pm 1$ .

Signs. $f'(-2)>0$ , $f'(0)<0$ , $f'(2)>0$ .

Classify. At $x=-1$ the sign changes from $+$ to $-$ , so $f$ has a local maximum there.

Classify. At $x=1$ the sign changes from $-$ to $+$ , so $f$ has a local minimum there.

Exam Variation (Context / Parameter Shift): A differentiable function $f$ has $f'(x)=(x-6)^{2}(x+2)$ . Does $f$ have a relative minimum, a relative maximum, or neither at $x=6$ ?

Solution: The factor $(x-6)^{2}$ is positive on both sides of $6$ , and $(x+2)$ is positive near $6$ , so $f'>0$ on both sides. There is no sign change, so $f$ has neither a relative minimum nor a relative maximum at $x=6$ . The critical number exists; the extreme does not.

Common Traps & Failure Signals:

The type without the evidence (Signal: "local max" with no sign statement): the conclusion was reported without the observation that supports it.
The value used as the type (Signal: reasoning from $f'(c)=0$ ): the derivative's value was mistaken for the derivative's sign pattern.
The absolute word (Signal: "maximum" unqualified): the local claim was upgraded to a global one the test never supports.

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

One Side Only: You checked the sign to the left of $c$ and stopped, so a change of sign could not be detected and every critical number looked like a maximum.
Type Assumed: You decided the type from the shape you expected rather than from the sign pattern, which works on textbook parabolas and fails on every function with a repeated root.
Word Upgraded: You wrote maximum for a local maximum.

Past 10-Year AP Exam Appearance

Testing Frequency: Most Years; Four released BC sets ask for a classification with a reason, and three of them supply the derivative or its graph rather than asking for it to be derived first
10-Year Total Count: 4 of 9 Exams · 4 FRQ Parts; Counted where the scored step is the classification of a critical point by a derivative test
FRQ Years Appeared: 2017 (Q5b), 2023 (Q4a), 2024 (Q4b), 2026 (Q2C)

Practice Examples

MCQ 1

A Simple Pair

For $f(x)=x^{3}-12x$ , what does the first derivative test give at $x=2$ ?

MCQ 2

A Squared Factor

The derivative of a differentiable function is $f'(x)=(x-3)^{2}(x+1)$ . What happens at $x=3$ ?

MCQ 3

At a Corner

For $f(x)=\lvert x-4\rvert$ , what does the first derivative test give at $x=4$ ?

FRQ 1 • Classify Every Candidate

Let $f(x)=x^{4}-8x^{2}+16$ .

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

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

(c) Determine the sign of $f'$ on each interval between consecutive critical numbers.

(d) Classify each critical number as a local maximum, a local minimum, or neither, with a reason for each.

Cheatsheet