# Determining Concavity

> AP Calculus BC · Analytical Applications of Differentiation
> Source: https://www.owlsprep.com/study/ap-calculus-bc-u5-determining-concavity/

This guide covers formal definitions of concave up/concave down, the second derivative rule for concavity, finding intervals of concavity, and identifying confirmed inflection points for AP Calculus BC, with exam-aligned worked examples.

**Prerequisites:** Computing first and second derivatives of explicit and implicit functions; Solving polynomial and rational inequalities to test sign changes; Finding domain of functions and evaluating function values

## Learning objectives

- Define concavity using first derivative behavior and second derivative sign
- Systematically find intervals of concavity for differentiable functions
- Correctly identify and confirm inflection points
- Write justifications for concavity that meet AP FRQ requirements

## Core Definition of Concavity and the Second Derivative Rule

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.

**Concavity** — Concavity describes the curvature of a function based on the behavior of its first derivative: a function is concave up if $f'$ is increasing on an interval, and concave down if $f'$ is decreasing on an interval. AP exclusively uses 'concave up' and 'concave down' terminology.

*Example:* The upward parabola $f(x)=x^2$ is concave up everywhere, while the downward parabola $f(x)=-x^2$ is concave down everywhere.

$$\begin{align*} f \text{ is concave up on } I &\iff f''(x) > 0 \text{ for all } x \in I \\ f \text{ is concave down on } I &\iff f''(x) < 0 \text{ for all } x \in I \end{align*}$$

**Worked example:** Use the second derivative rule to confirm that $f(x) = 2e^{2x} - 12x$ is concave up over its entire domain.

1. Compute the first derivative:

   $$f'(x) = 4e^{2x} - 12$$
2. Compute the second derivative:

   $$f''(x) = 8e^{2x}$$
3. Check the sign of $f''(x)$: For all real $x$, $e^{2x} > 0$, so $8e^{2x} > 0$ for all $x$ in the domain of $f$.
4. By the second derivative rule, since $f''(x)$ is positive everywhere, $f(x)$ is concave up over its entire domain.

> **tip**
>
> On AP FRQs, you must explicitly reference the sign of the second derivative to justify concavity. A conclusion of 'concave up' without this reference will lose points.

## Finding Intervals of Concavity

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.

1. Compute $f''(x)$ fully
2. Find all candidate points: all $x$ in the domain of $f$ where $f''(x) = 0$ or $f''(x)$ is undefined
3. Candidate points split the domain of $f$ into open test intervals
4. Test the sign of $f''(x)$ in each interval: positive = concave up, negative = concave down

**Worked example:** Find all intervals of concavity for $f(x) = \frac{x^3}{3} - x^2 - 3x + 2$.

1. Compute derivatives:

   $$f'(x) = x^2 - 2x - 3 \implies f''(x) = 2x - 2 = 2(x - 1)$$
2. Find candidate points: $f''(x)$ is defined everywhere for all real $x$. Set $f''(x)=0$, giving $x=1$. This splits the domain into two test intervals: $(-\infty, 1)$ and $(1, \infty)$.
3. Test the sign of $f''$: For $(-\infty, 1)$, test $x=0$: $f''(0) = -2 < 0$. For $(1, \infty)$, test $x=2$: $f''(2) = 2 > 0$.
4. Assign concavity: $f(x)$ is concave down on $(-\infty, 1)$ and concave up on $(1, \infty)$.

> **tip**
>
> Never forget to check for points where $f''(x)$ is undefined (but $x$ is in the domain of $f$). Most students only check for $f''(x)=0$ and miss concavity changes at corners, cusps, or vertical tangents.

## Identifying and Confirming Inflection Points

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 $f$.

**Inflection Point** — A point $(c, f(c))$ on the graph of $f$ where the concavity of $f$ changes from up to down, or down to up, across $x=c$. Requires that $c$ is in the domain of $f$ and $f''$ changes sign across $x=c$.

**Worked example:** Find all inflection points of $f(x) = x^4 - 8x^3 + 18x^2$.

1. Compute the second derivative:

   $$f'(x) = 4x^3 - 24x^2 + 36x \implies f''(x) = 12x^2 - 48x + 36 = 12(x-1)(x-3)$$
2. Find candidate points: $f''$ is defined everywhere, set equal to zero to get $x=1$ and $x=3$, both in the domain of $f$.
3. Check for sign change: Test $(-\infty, 1)$: $f''(-1) = 96 > 0$ (concave up). For $(1,3)$: $f''(2) = -12 < 0$ (concave down). For $(3, \infty)$: $f''(4) = 36 > 0$ (concave up). Sign changes at both $x=1$ and $x=3$, so both are inflection points.
4. Compute $y$-coordinates: $f(1) = 11$, $f(3) = 27$, so inflection points are $(1, 11)$ and $(3, 27)$.

> **tip**
>
> AP FRQs require inflection points to be written as ordered pairs $(x, y)$, not just $x$-coordinates. Always compute the $y$-value to earn full credit.

## AP Style Concept Check

**Check your understanding**

Test your understanding with these AP-style questions:

1. How many distinct intervals of concave up does $f(x) = x^4 - 6x^2 + 8x + 1$ have over all real numbers?

   - 1
   - 2
   - 3
   - 4

   *Why:* Correct: $f''(x) = 12(x-1)(x+1)$, which is positive on two intervals: $(-\infty, -1)$ and $(1, \infty)$.

2. Which of the following is a required condition for $(c, f(c))$ to be an inflection point?

   - $f''(c) = 0$
   - The sign of $f''$ changes across $x=c$
   - $f'(c) = 0$
   - $f''(c)$ is undefined

   *Why:* Correct: $f''(c) = 0$ only makes $x=c$ a candidate; a sign change is required for a concavity change.

## Common pitfalls

- **Wrong:** Claiming $f''(c) = 0$ means $(c, f(c))$ must be an inflection point
  - Why it fails: Students confuse a common property of inflection points with a requirement, forgetting that a sign change is mandatory
  - Correct: Always test whether the sign of $f''$ changes across $x=c$ before concluding it is an inflection point
- **Wrong:** Ignoring points where $f''(x)$ is undefined (but $x$ is in the domain of $f$) when searching for inflection points
  - Why it fails: Students only search for roots of $f''(x) = 0$ and forget $f''$ can be undefined at points on $f$ where concavity changes
  - Correct: Always list all $x$ in the domain of $f$ where $f''$ is zero or undefined before dividing into test intervals
- **Wrong:** Justifying concavity by referencing the sign of the first derivative, not the second
  - Why it fails: Students confuse the test for increasing/decrease (first derivative) with the test for concavity (second derivative)
  - Correct: Explicitly reference the sign of the second derivative in all FRQ justifications for concavity
- **Wrong:** Reporting an inflection point at an $x$-value not in the domain of the original function
  - Why it fails: Students find a root of $f''$ but forget to check if the original function is defined there
  - Correct: Always confirm that $x=c$ is in the domain of $f$ before checking for an inflection point
- **Wrong:** Closing intervals of concavity by including inflection point $x$-values
  - Why it fails: Concavity is defined for open intervals, not individual points
  - Correct: Always use open intervals when reporting intervals of concavity, as required by the AP exam

## Cheatsheet

| Category | Rule / Key Statement | AP Exam Notes |
| --- | --- | --- |
| Concave up on $I$ | $f'$ increasing $\iff f''(x) > 0$ for all $x \in I$ | Graph lies above tangents; test with $f''$ sign |
| Concave down on $I$ | $f'$ decreasing $\iff f''(x) < 0$ for all $x \in I$ | Graph lies below tangents; always use open intervals |
| Candidate concavity change points | All $x$ in domain of $f$ where $f''(x)=0$ or $f''$ undefined | Candidates are not automatically inflection points; need sign change |
| Inflection point requirement | $(c, f(c))$ where concavity changes across $x=c$ | Requires $c$ in domain of $f$ + sign change of $f''$ |
| Process for intervals of concavity | 1. Compute $f''$ 2. Find candidates 3. Test sign 4. Assign concavity | Never skip checking for undefined $f''(x)$ in $f$'s domain |
| Connection to first derivative | Concave up $\iff f'$ increasing; concave down $\iff f'$ decreasing | Used to test if a rate of change is increasing/decreasing |
| AP FRQ Justification | "$f$ is concave up on $I$ because $f''(x) > 0$ for all $x \in I$" | No reference to $f''$ sign = no points |
| Reporting inflection points | Write as ordered pair $(c, f(c))$ | Only reporting $x$-coordinate loses points |

## What's next

Determining concavity is a direct prerequisite for the second derivative test for local extrema, which you will apply next to classify critical points as local minima or maxima. Without a solid understanding of how to compute the second derivative and test its sign, you cannot correctly apply this common test, which appears across both MCQ and FRQ sections of the AP exam. Beyond extrema classification, concavity is a core tool for full analytic curve sketching, a frequent multi-part FRQ task. It also appears in applied problems from kinematics to economics, so mastery is critical for all applied derivative questions on the exam.

- [Second Derivative Test](https://www.owlsprep.com/study/ap-calculus-bc-u5-second-derivative-test/)
- [Sketching graphs of f, f', f''](https://www.owlsprep.com/study/ap-calculus-bc-u5-sketching-graphs-of-f-f/)
- [Connecting f, f', f'' qualitatively](https://www.owlsprep.com/study/ap-calculus-bc-u5-connecting-f-f-f-qualitatively/)

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