Calculus BC · Analytical Applications of Differentiation · 14 min read · Updated 2026-05-11
Determining Concavity — AP Calculus BC
AP Calculus BC · Analytical Applications of Differentiation · 14 min read
1. Core Definition of Concavity and the Second Derivative Rule★★☆☆☆⏱ 4 min
Concavity describes the direction of curvature of a function's graph, relative to its tangent lines across an interval. This topic makes up 4-6% of the AP Calculus BC exam, appearing in both multiple choice and free response, almost always paired with curve sketching or extrema classification.
\begin{align*} f \text{ is concave up on } I &\iff f''(x) > 0 \text{ for all } x \in I \\ f \text{ is concave down on } I &\iff f''(x) < 0 \text{ for all } x \in I \end{align*}
2. Finding Intervals of Concavity★★★☆☆⏱ 4 min
To find intervals of concavity, we use a systematic process that mirrors finding intervals of increase/decrease, but uses the second derivative. Concavity can only change at candidate points: points in the domain of $f$ where $f''(x) = 0$ or $f''(x)$ is undefined.
Compute $f''(x)$ fully
Find all candidate points: all $x$ in the domain of $f$ where $f''(x) = 0$ or $f''(x)$ is undefined
Candidate points split the domain of $f$ into open test intervals
Test the sign of $f''(x)$ in each interval: positive = concave up, negative = concave down
3. Identifying and Confirming Inflection Points★★★☆☆⏱ 3 min
An inflection point is a point on the graph of $f$ where concavity changes. A common misconception is that all points where $f''(c) = 0$ are inflection points, which is not true: only points where the sign of $f''$ changes are inflection points, even if $f''$ is undefined there, as long as $c$ is in the domain of $f$.
4. AP Style Concept Check★★★★☆⏱ 3 min
Common Pitfalls
Why: Students confuse a common property of inflection points with a requirement, forgetting that a sign change is mandatory
Why: Students only search for roots of $f''(x) = 0$ and forget $f''$ can be undefined at points on $f$ where concavity changes
Why: Students confuse the test for increasing/decrease (first derivative) with the test for concavity (second derivative)
Why: Students find a root of $f''$ but forget to check if the original function is defined there
Why: Concavity is defined for open intervals, not individual points