Second Derivative Test — AP Calculus AB
1. Core Definition and Purpose ★★☆☆☆ ⏱ 3 min
The second derivative test is an analytical method to classify critical points of a twice-differentiable function as local maxima, local minima, or inconclusive. For AP Calculus AB, this topic falls in Unit 5, which makes up 15–18% of your total exam score, and appears in both multiple-choice (MCQ) and free-response (FRQ) sections.
Unlike the first derivative test, which requires analyzing the sign change of $f'(x)$ around a critical point, the second derivative test only requires evaluating the second derivative at the critical point itself, making it faster when the second derivative is easy to compute. It has clear limitations you must recognize for the exam.
2. Formal Statement and Geometric Intuition ★★☆☆☆ ⏱ 4 min
A critical point of $f(x)$ is a point $x=c$ in the domain of $f$ where $f'(c) = 0$ or $f'(c)$ is undefined. The second derivative test only applies to critical points where $f'(c) = 0$ and $f''(c)$ exists; if $f'(c)$ is undefined, you must use the first derivative test regardless.
The intuition comes from concavity: if $f''(c) < 0$, the function is concave down at $c$, curving downward around the horizontal tangent to form a peak (local maximum). If $f''(c) > 0$, the function is concave up, curving upward around the horizontal tangent to form a valley (local minimum).
Exam tip: Always explicitly connect the sign of $f''(c)$ to your classification to earn full credit on FRQs.
3. Inconclusive Cases and Resolution ★★★☆☆ ⏱ 3 min
When $f''(c) = 0$ at a critical point where $f'(c)=0$, the second derivative test cannot give a definitive answer. This does not mean there is no extremum at $x=c$: for example, $f(x)=x^4$ has $f''(0)=0$ but $x=0$ is a local minimum, while $f(x)=x^3$ also has $f''(0)=0$ but $x=0$ is not an extremum.
When the test is inconclusive, the only valid approach on the AP exam is to fall back to the first derivative test: check the sign of $f'(x)$ on intervals on either side of $c$.
Exam tip: Leaving an inconclusive point unclassified will cost you at least one point on FRQs.
4. Second Derivative Test for Absolute Extrema ★★★☆☆ ⏱ 4 min
By the Extreme Value Theorem, absolute extrema of a continuous function on a closed interval $[a,b]$ occur either at critical points inside the interval or at the endpoints. After using the second derivative test to classify local extrema, you still must compare function values at all candidates to find the absolute maximum and minimum.
This workflow is faster than using the first derivative test for classification, and is the most common approach for optimization FRQs on the AP exam.
Exam tip: Never apply the second derivative test to endpoints, they are only candidates for absolute extrema.
5. Check Your Understanding ★★★☆☆ ⏱ 3 min
Common Pitfalls
Why: The test requires $f'(c)=0$ and $f''(c)$ exists to work, and critical points can occur where $f'(c)$ is undefined.
Why: Inconclusive means the test cannot give a result, not that there is no extremum. Many extrema have $f''(c)=0$.
Why: Confusion between concavity direction and the type of extremum is a very common exam mistake.
Why: Students confuse critical points (found from first derivative) with inflection points (found from second derivative).
Why: Students assume the absolute maximum must be the largest local maximum, but the Extreme Value Theorem allows absolute extrema at endpoints.