Calculus BC · Unit 5: Analytical Applications of Differentiation · 14 min read · Updated 2026-05-11
First derivative test for relative extrema — AP Calculus BC
AP Calculus BC · Unit 5: Analytical Applications of Differentiation · 14 min read
1. Core Concepts of the First Derivative Test★★☆☆☆⏱ 3 min
The First Derivative Test (also called the test for local/relative extrema) is a core method from AP Calculus BC Unit 5, appearing on both multiple-choice and free-response sections, accounting for 4-7% of total exam score.
The test uses the geometric meaning of the first derivative: $f'(x)$ tells us if $f(x)$ is increasing ($f'(x) > 0$) or decreasing ($f'(x) < 0$). By analyzing how the sign of $f'(x)$ changes around a critical point, we can classify if the point is a relative maximum, minimum, or neither.
Exam tip: On AP FRQs, you must explicitly mention the sign change of the first derivative to earn full justification credit for classifying extrema.
2. Sign Analysis for Interior Critical Points★★☆☆☆⏱ 4 min
To apply the First Derivative Test to an interior critical point, split the domain of $f'$ into intervals separated by critical points, test the sign of $f'(x)$ at a point inside each interval, then use the sign change rules below to classify:
If $f'$ changes from positive to negative at $c$, $f$ has a **relative maximum** at $x=c$
If $f'$ changes from negative to positive at $c$, $f$ has a **relative minimum** at $x=c$
If $f'$ does not change sign at $c$, $f$ has **no relative extremum** at $x=c$
Exam tip: Stating only that $f'(c)=0$ is not sufficient justification for an extremum on FRQs.
3. Critical Points Where $f'$ is Undefined★★★☆☆⏱ 4 min
Critical points can occur where $f'(c)$ is undefined, as long as $c$ is in the domain of $f$. The First Derivative Test follows exactly the same process for these points as it does for points where $f'(c)=0$. This scenario is common for absolute value, root, and piecewise functions, which appear regularly on AP exams.
Exam tip: Always confirm that the point where $f'$ is undefined is in the domain of $f$ before calling it a critical point.
4. First Derivative Test for Endpoint Extrema★★★☆☆⏱ 4 min
On a closed interval $[a, b]$, endpoints $x=a$ and $x=b$ can be relative extrema, since we only consider function values inside the interval when defining relative extrema. The First Derivative Test extends naturally to endpoints by only checking the sign of $f'$ on the interior side of the endpoint:
Left endpoint $x=a$: If $f'(x) < 0$ just right of $a$, $f(a)$ is a relative maximum. If $f'(x) > 0$ just right of $a$, $f(a)$ is a relative minimum.
Right endpoint $x=b$: If $f'(x) > 0$ just left of $b$, $f(b)$ is a relative maximum. If $f'(x) < 0$ just left of $b$, $f(b)$ is a relative minimum.
Exam tip: When asked to find all relative extrema on a closed interval, don't forget to classify endpoints—this is a common AP exam point trap.
5. AP-Style Additional Worked Examples★★★★☆⏱ 6 min
Common Pitfalls
Why: Students automatically mark any point where $f'$ is undefined as a critical point, without checking the domain of $f$.
Why: Students confuse the condition for a critical point with the justification for an extremum.
Why: Students are taught critical points are interior points, so they ignore endpoints entirely when classifying relative extrema.
Why: Students sometimes accidentally test a critical point instead of a point inside the interval, getting a zero derivative and incorrectly concluding no sign change.
Why: Students mix up function and derivative values when working quickly on exam day.