First Derivative Test for Relative Extrema — AP Calculus AB
1. What is the First Derivative Test for Relative Extrema? ★★☆☆☆ ⏱ 3 min
The First Derivative Test is a core technique in Unit 5 of AP Calculus AB, which makes up 15–18% of your total exam score. It is tested in both multiple-choice and free-response questions, often paired with curve sketching and optimization. The test uses the sign change of the first derivative around a critical point to classify the point.
2. Critical Points and Sign Interval Setup ★★☆☆☆ ⏱ 3 min
Before applying the First Derivative Test, you must complete two preliminary steps: find all critical points, then split the domain into intervals separated by critical points to test the sign of $f'(x)$. Fermat's Theorem confirms all relative extrema occur at critical points, so no other locations need testing.
Exam tip: Always sort your critical points from smallest to largest before creating intervals. Skipping this step often leads to testing the wrong interval and misclassifying extrema on the AP exam.
3. First Derivative Test Classification Rules ★★☆☆☆ ⏱ 3 min
The classification rules follow directly from the relationship between derivative sign and function behavior: a positive first derivative means the function is increasing, and a negative first derivative means the function is decreasing. The sign change pattern as you move left to right across $x=c$ tells you the classification:
- If $f'(x)$ changes from **positive to negative**: $f(c)$ is a **relative maximum**
- If $f'(x)$ changes from **negative to positive**: $f(c)$ is a **relative minimum**
- If $f'(x)$ does not change sign: there is **no relative extremum** at $x=c$
- These rules work for all critical points, whether $f'(c)=0$ or $f'(c)$ is undefined
Exam tip: Always compute the full function value for the extremum if the question asks for the point or value. The AP exam will dock points if you only give the x-coordinate when the question asks for the full extremum.
4. Classifying Critical Points with Undefined Derivatives ★★★☆☆ ⏱ 3 min
The First Derivative Test works exactly the same for critical points where $f'(x)$ is undefined (corner points, cusps, vertical tangents), as long as the original function $f(x)$ is defined at the critical point. A common mistake is omitting these critical points from your list, which AP exam writers regularly test for.
Exam tip: Never forget to add critical points where $f'(x)$ is undefined to your list. AP exam writers regularly include these to test if you remember all types of critical points.
5. AP-Style Concept Check ★★★☆☆ ⏱ 2 min
Common Pitfalls
Why: Students confuse 'possible location of an extremum' with 'guaranteed extremum', as most introductory examples have extrema at every stationary point.
Why: Students only look for solutions to $f'(x)=0$ and ignore zeros in the denominator of the derivative.
Why: Students mix up the direction when moving left to right across the critical point.
Why: Students rush and pick the interval endpoint as the test point, which gives $f'(c)=0$ or undefined, so no valid sign.
Why: Students forget you need derivative values on both sides of the point to check for a sign change.
Why: Students approximate irrational test values and round a small negative number to zero or a positive number.