Determining Concavity — AP Calculus AB
1. Core Definition of Concavity ★★☆☆☆ ⏱ 3 min
Concavity describes the direction of curvature of a differentiable function $f(x)$ over an interval, and it is a core high-frequency skill on the AP Calculus AB exam. Unit 5 (Analytical Applications of Differentiation) makes up 10–15% of total exam weight, and concavity questions appear in both multiple-choice and free-response sections.
Informally, concave up intervals are shaped like a cup ($\cup$) that "holds water", and concave down intervals are shaped like a cap ($\cap$) that "spills water".
2. Second Derivative Rule & Interval Testing ★★★☆☆ ⏱ 4 min
Concavity describes how the slope of the tangent line (the first derivative $f'(x)$) changes as $x$ increases. If a function is concave up, its slope increases as $x$ increases; if it is concave down, its slope decreases as $x$ increases. Because the second derivative $f''(x)$ measures the rate of change of $f'(x)$, this gives a direct testable relationship between the sign of $f''$ and concavity.
- Compute the second derivative $f''(x)$
- Find all candidate split points where $f''(x) = 0$ or $f''(x)$ is undefined (these split the domain into open test intervals)
- Test the sign of $f''(x)$ in each interval
- Assign concavity based on the sign of $f''(x)$
3. Identifying Inflection Points ★★★☆☆ ⏱ 3 min
An inflection point is a point on the graph of $f$ where concavity changes from up to down or down to up. For an inflection point to exist at $x=c$, two non-negotiable conditions must be met: (1) $f(c)$ is defined (the point $(c, f(c))$ lies on the graph of $f$), and (2) the sign of $f''(x)$ (and thus concavity) changes across $x=c$.
A common student misconception is that inflection points only occur where $f''(c)=0$, and that all points with $f''(c)=0$ are inflection points. This is incorrect: inflection points can also occur where $f''(c)$ is undefined (as long as $f(c)$ exists and concavity changes), and $f''(c)=0$ does not guarantee a concavity change.
4. Concavity from a Graph of $f'(x)$ ★★★★☆ ⏱ 4 min
The AP exam frequently tests the skill of determining concavity when given only the graph of the first derivative $f'(x)$, not an explicit formula for $f(x)$. We use the core relationship: $f''(x)$ is equal to the slope of the tangent line to the graph of $f'(x)$ at $x$.
This gives us a simple rule: if $f'(x)$ is increasing on an interval, its slope is positive, so $f''(x) > 0$, so $f(x)$ is concave up. If $f'(x)$ is decreasing on an interval, its slope is negative, so $f''(x) < 0$, so $f(x)$ is concave down. Inflection points on $f(x)$ correspond exactly to local extrema (peaks or valleys) on the graph of $f'(x)$.
Common Pitfalls
Why: Students memorize that inflection points occur where $f''=0$, so they stop there and skip the required sign check
Why: Students mix up the two uses of the first derivative, one for monotonicity of $f$ and one for concavity of $f$ via the slope of $f'$
Why: Students assume endpoints should be included because $f$ is defined there, but concavity is defined for open intervals
Why: Students only look for where $f''(x)=0$ and forget that points where the second derivative does not exist can still be inflection points
Why: Students incorrectly associate 'decreasing' with 'concave down', confusing slope direction with curvature direction