Determining Concavity of Functions over Their Domains
Use changing slope to determine concavity and verify an actual concavity change at a point of the graph.
College Board Topic 5.6 · CED p. 99 (PDF p. 104) · CED effective Fall 2020; project course outline 2026–27.
Concavity is about how slope changes. If f′ increases on an interval, f is concave up there; if f′ decreases, f is concave down. Where f″ exists, f″ > 0 is a useful test for upward concavity and f″ < 0 for downward concavity. So a decreasing function can be concave up: its negative slopes may become less negative.
An inflection point is a point of the function where concavity changes. A zero or undefined value of f″ is a candidate, not a finished answer. Check the signs on nearby intervals and make sure the original function has a point there with the required continuity. The function x⁴ has f″(0) = 0 but stays concave up, so zero is not an inflection point. The function 1/x changes second-derivative sign across zero, but zero is outside its domain, so it is not an inflection point.
A second derivative need not exist at an inflection point. The continuous function x^(1/3) is concave up for x < 0 and concave down for x > 0, so its graph has an inflection point at the origin. Its vertical tangent also makes f′ undefined there. Use the concavity-change definition rather than a rule that blindly solves f″ = 0.
| Evidence | Conclusion about f |
|---|---|
| f′ > 0 | Increasing |
| f′ < 0 | Decreasing |
| f′ increasing, or f″ > 0 | Concave up |
| f′ decreasing, or f″ < 0 | Concave down |
Let's work through it
For f(x) = x³ − 6x², f′ = 3x² − 12x and f″ = 6x − 12.
Why: Differentiate twice to test how the slope changes.
f″ < 0 for x < 2 and f″ > 0 for x > 2.
Why: Solve the sign inequality, not just the zero equation.
f is concave down on (−∞,2) and up on (2,∞).
Why: Negative/positive second derivative means decreasing/increasing slope.
There is an inflection point at (2,−16).
Why: The polynomial is continuous and the concavity changes; evaluate f(2) to locate the point.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
Must f″ = 0 give an inflection point?
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
Must f″ = 0 give an inflection point?
Worked answer and reason
No; verify a concavity change.
x⁴ at zero is a counterexample.
Retrieval 2
Can f decrease while concave up?
Worked answer and reason
Yes, if f′ < 0 while f′ is increasing.
Slope sign and slope change are different.
Retrieval 3
Is zero an inflection point of 1/x?
Worked answer and reason
No, it is outside the domain.
An inflection point must be a point of the graph.
Catch the tempting mistake
- Concave up means the function is positive.
- Concavity describes changing slopes, independently of height.
- Every second-derivative sign change across a number is an inflection point.
- The function must be defined and have the appropriate continuous point there.
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Put it to work
- Unit 5 problem 12 · No calculator · analytic
- Unit 5 problem 13 · No calculator · analytic
- Unit 5 problem 29 · No calculator · multiple
For the teacher
- Ask for signs of both f′ and f″ in one sentence: “decreasing and concave up.”
- Use x⁴, x^(1/3), and 1/x as a comparison set so the definition survives beyond polynomial examples.
Glossary and deeper reading · 6 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Concavity
Concavity describes how a graph’s slope changes as the input increases. A graph is concave up where its derivative increases and concave down where its derivative decreases. For example, x² is concave up because its derivative 2x increases. A positive second derivative establishes concave-up behavior on an interval, while a negative one establishes concave-down behavior.
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Interval
An interval is a set of real numbers containing every number between any two of its members. Parentheses exclude an endpoint, while square brackets include it. For example, (1,4] includes 4 and every number between 1 and 4, but excludes 1. An infinite endpoint always uses a parenthesis because infinity is not a real endpoint.
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Inflection point
An inflection point is a point on a continuous graph where its concavity changes. A zero or undefined second derivative may identify a candidate but does not establish that change. For example, x³ changes concavity at 0, while x⁴ has second derivative zero there without changing concavity. Check the behavior on both sides of the candidate.
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Function
A mathematical function assigns exactly one output to each input in its domain. Inputs can share an output, but one input cannot have two different outputs. For example, f(x)=x² assigns the output 9 to both 3 and −3. Function notation names the rule and its input; f(3) means the output when the input is 3.
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Second derivative
A second derivative is the derivative of the first derivative. It measures how the first rate of change changes with the input; further repeated differentiation gives higher-order derivatives. For example, f(x)=x³ has f′(x)=3x² and f″(x)=6x. For position measured against time, the second derivative is acceleration.
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Continuity
A function is continuous at an input when its value exists, its finite limit exists, and those two values agree. At a domain endpoint, continuity uses the relevant one-sided approach. For example, x² is continuous at 3 because both its value and its nearby limit are 9. Continuity on an interval requires the appropriate condition at every input in that interval.