AP® Calculus AB review sheet from Aim for Five (aimforfive.com/calc-ab/units/5/5-6)
Unit 5 · Topic 5.6
5.6 Determining Concavity of Functions over Their Domains
Concavity describes how a graph bends. It's concave up (like a cup) where f′ is increasing, which happens where f″ > 0, and concave down where f′ is decreasing, where f″ < 0. A point of inflection is where the concavity actually changes.
Key terms
- concave up
- concave down
- second derivative
- point of inflection
Concave up and concave down
f is concave up on an interval if f′ is increasing there. The slopes get bigger as you move right, so the graph bends upward like a cup. f is concave down if f′ is decreasing: the slopes get smaller, and the graph bends downward like a cap.
Concavity is separate from increasing and decreasing. A graph can rise while concave down (rising more and more slowly) or fall while concave up (falling more and more slowly).
In context, concavity describes how a rate is changing. If the height of a plant is concave up over time, the plant is growing faster and faster; if it's concave down, its growth is slowing.
The second derivative tells you
Since f″ is the derivative of f′, its sign tells you whether f′ is increasing:
- f″ > 0 on an interval means f′ is increasing, so f is concave up.
- f″ < 0 on an interval means f′ is decreasing, so f is concave down.
- To find intervals of concavity, find where f″ = 0 or is undefined, then make a sign chart for f″, just as you did for f′ in 5.3.
Points of inflection
A point of inflection is a point on the graph where f is continuous and the concavity changes, from up to down or down to up. So f″ must change sign there (or, using f′, f′ changes from increasing to decreasing or the reverse).
f″ = 0 is not enough. For f(x) = x⁴, f″(x) = 12x² is 0 at x = 0 but positive on both sides, so the graph is concave up the whole time and there's no inflection point. Inflection points can also occur where f″ doesn't exist, as long as the sign of f″ changes and f is continuous there.
Justification language
For concavity: “f is concave up on (2, ∞) because f″(x) > 0 there.” With a graph of f′: “because f′ is increasing on (2, ∞).”
For inflection: “f has a point of inflection at x = 2 because f″ changes sign at x = 2.” With a graph of f′: “because f′ changes from decreasing to increasing at x = 2,” which on the graph of f′ looks like a valley (relative min of f′).
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Concavity and inflection of a cubic
For f(x) = x³ − 6x² + 5, find the intervals of concavity and the point of inflection.
Show the solutionHide the solution
- Step 1: f′(x) = 3x² − 12x and f″(x) = 6x − 12 = 6(x − 2).
- Step 2: f″ < 0 for x < 2 and f″ > 0 for x > 2.
- Step 3: f″ changes sign at x = 2, and f(2) = 8 − 24 + 5 = −11.
Answer: Concave down on (−∞, 2), concave up on (2, ∞). Point of inflection at (2, −11), because f″ changes sign there.
- Example 2
Concavity from a graph of f′
The graph of f′ on [0, 6] rises from (0, −1) to a peak at (2, 3), then falls to (6, −2). Where is f concave up, where is it concave down, and where does f have an inflection point?
Show the solutionHide the solution
- Step 1: Concavity of f depends on whether f′ is increasing or decreasing, not on its sign.
- Step 2: f′ is increasing on (0, 2), so f is concave up there.
- Step 3: f′ is decreasing on (2, 6), so f is concave down there.
- Step 4: At x = 2, f′ switches from increasing to decreasing, so the concavity of f changes.
Answer: f is concave up on (0, 2), concave down on (2, 6), and has a point of inflection at x = 2.
- Example 3
Trap: f″ = 0 without a sign change
Does f(x) = x⁴ − 4x have a point of inflection at x = 0?
Show the solutionHide the solution
- Step 1: f′(x) = 4x³ − 4 and f″(x) = 12x².
- Step 2: f″(0) = 0, which makes x = 0 a candidate.
- Step 3: But f″(x) = 12x² > 0 for x < 0 and for x > 0. The sign doesn't change.
Answer: No. f″(0) = 0, but f″ is positive on both sides, so f is concave up throughout and has no inflection point at x = 0.
Common mistakes
- Saying there's an inflection point wherever f″ = 0. The sign of f″ must change.
- Using the sign of f′ to decide concavity. Concavity depends on whether f′ is increasing or decreasing.
- On a graph of f′, looking for x-intercepts to find inflection points. Look for peaks and valleys of f′ instead.
On the exam
- Free-response questions often ask for inflection points of f from a graph of f′. Point to where f′ changes from increasing to decreasing (or the reverse).
- Concavity also justifies whether a tangent line approximation is an overestimate or underestimate (4.6).
Connected topics
Videos
Check yourself
4 questions on 5.6 Determining Concavity of Functions over Their Domains. Pick an answer to see if you got it, and why.
At which values of x does the graph of f(x) = x⁵ − 5x⁴ have a point of inflection?
The derivative of a function f is given by f′(x) = 2 sin x − 0.1x³. On the interval 0 < x < 3, what is the x-coordinate of the point of inflection of the graph of f?
On which of the following intervals is the graph of f(x) = ln(x² + 1) concave up?
A function f is twice differentiable for all real x, with f(0) = 1, f′(x) > 0 for all x and f″(x) < 0 for all x. Which of the following could be f(x) ?
0 of 4 answered