How a curve bends
Concavity describes the direction a graph curves.
| Concavity | Shape | f″(x) | Slopes |
|---|---|---|---|
| Concave up | ∪ (holds water) | > 0 | increasing |
| Concave down | ∩ (spills water) | < 0 | decreasing |
Concavity is about the rate of change of slope, not whether the function is rising or falling. A graph can be increasing and concave down (rising but levelling off).
Points of inflection
A point of inflection is where the concavity changes, from up to down or down to up.
To find them:
- Find the second derivative f″(x).
- Solve f″(x) = 0 (or find where it’s undefined).
- Check that f″ changes sign there.
Example
f(x) = x³ − 6x² + 9x + 2
- f′(x) = 3x² − 12x + 9
- f″(x) = 6x − 12 = 0 → x = 2
| Interval | Test | f″ | Concavity |
|---|---|---|---|
| x < 2 | 0 | −12 | down |
| x > 2 | 3 | 6 | up |
Sign changes → point of inflection at x = 2: f(2) = 8 − 24 + 18 + 2 = (2, 4).
f″ = 0 isn’t enough
f(x) = x⁴: f″(x) = 12x² = 0 at x = 0, but f″ ≥ 0 on both sides. No sign change → not a point of inflection (the graph is concave up throughout).
Reading concavity in context
| Situation | Concavity meaning |
|---|---|
| sales increasing, concave down | growth is slowing |
| cost increasing, concave up | costs rising faster and faster |
| spread of an illness | inflection point = when new cases peak and start to slow |
The inflection point in an S-shaped (logistic) growth curve marks the moment of fastest growth.
Concavity and the second derivative test
- At a critical point in a concave up region → local min.
- In a concave down region → local max.
Worked examples
Example 1: f(x) = x² + 3x. Concavity? f″ = 2 > 0 → concave up everywhere.
Example 2: f(x) = −x³. Inflection point? f″ = −6x = 0 at x = 0; sign changes → (0, 0).
Example 3: Where is f(x) = x³ − 3x² concave down? f″ = 6x − 6 < 0 → x < 1.
Common mistakes
- Confusing increasing with concave up.
- Declaring an inflection point without a sign change.
- Using f′ instead of f″.
Try it yourself
- Concavity of f(x) = −2x²?
- Inflection point of f(x) = x³ + 3x² − 1?
- Where is f(x) = x⁴ − 6x² concave up?
- Is x = 0 an inflection point of f(x) = x⁴?
Answers: 1) concave down 2) (−1, 1) 3) x < −1 or x > 1 4) no
Quick recap
- Concavity: concave up (f″ > 0) or concave down (f″ < 0).
- A point of inflection is where the concavity changes.
- Check that the second derivative changes sign.