Determining Concavity

AP Calculus BC· AP Calculus BC CED — 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.

📘 Definition

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 $f (x) = 2 e^{2 x} - 12 x$ 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 $x$ , $e^{2 x} > 0$ , so $8 e^{2 x} > 0$ 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.

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

📐 Worked Example

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

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 $(1, \infty)$ .

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 $f$ .

📘 Definition

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 $f (x) = x^{4} - 8 x^{3} + 18 x^{2}$ .

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 $(3, 27)$ .

4. AP Style Concept Check★★★★☆⏱ 3 min

✓ Quick check

Test your understanding with these AP-style questions:

How many distinct intervals of concave up does $f (x) = x^{4} - 6 x^{2} + 8 x + 1$ have over all real numbers?
- 1
- 2
- 3
- 4
Reveal answer
2 —
Correct: $f^{''} (x) = 12 (x - 1) (x + 1)$ , which is positive on two intervals: $(- \infty, - 1)$ and $(1, \infty)$ .
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
Reveal answer
The sign of $f''$ changes across $x=c$ —
Correct: $f^{''} (c) = 0$ only makes $x = c$ a candidate; a sign change is required for a concavity change.

5. Common Pitfalls

Wrong move:

Claiming $f^{''} (c) = 0$ means $(c, f (c))$ must be an inflection point

Why:

Students confuse a common property of inflection points with a requirement, forgetting that a sign change is mandatory

Correct move:

Always test whether the sign of $f^{''}$ changes across $x = c$ before concluding it is an inflection point

Wrong move:

Ignoring points where $f^{''} (x)$ is undefined (but $x$ is in the domain of $f$ ) when searching for inflection points

Why:

Students only search for roots of $f^{''} (x) = 0$ and forget $f^{''}$ can be undefined at points on $f$ where concavity changes

Correct move:

Always list all $x$ in the domain of $f$ where $f^{''}$ is zero or undefined before dividing into test intervals

Wrong move:

Justifying concavity by referencing the sign of the first derivative, not the second

Why:

Students confuse the test for increasing/decrease (first derivative) with the test for concavity (second derivative)

Correct move:

Explicitly reference the sign of the second derivative in all FRQ justifications for concavity

Wrong move:

Reporting an inflection point at an $x$ -value not in the domain of the original function

Why:

Students find a root of $f^{''}$ but forget to check if the original function is defined there

Correct move:

Always confirm that $x = c$ is in the domain of $f$ before checking for an inflection point

Wrong move:

Closing intervals of concavity by including inflection point $x$ -values

Why:

Concavity is defined for open intervals, not individual points

Correct move:

Always use open intervals when reporting intervals of concavity, as required by the AP exam

6. Quick Reference Cheatsheet

Category	Rule / Key Statement	AP Exam Notes
Concave up on $I$	$f^{'}$ increasing $⟺ f^{''} (x) > 0$ for all $x \in I$	Graph lies above tangents; test with $f^{''}$ sign
Concave down on $I$	$f^{'}$ decreasing $⟺ f^{''} (x) < 0$ 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	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 $⟺ f^{'}$ increasing; concave down $⟺ f^{'}$ 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

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2023 · BC
Find inflection points of a function
2022 · BC FRQ
Justify concavity in applied context
2021 · BC
Find intervals of concavity

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.