5.6 Concavity

Sign of f''

f is concave up where f''(x) > 0 (shaped like a cup; f' is increasing) and concave down where f''(x) < 0 (f' is decreasing).

Inflection points

An inflection point is where concavity changes — that is, where f'' changes sign (f'' is 0 or undefined there and switches sign).

Key terms

Concave up

f''(x) > 0; f' increasing.

Concave down

f''(x) < 0; f' decreasing.

Inflection point

Where f'' changes sign (concavity changes).

Concavity & f'

Concave up ⇔ f' increasing.

Second derivative

Its sign gives concavity.

Try an AP-style question

Where is f(x) = x³ − 3x² concave up?

1. x > 2
2. x < 1
3. x > 1
4. 0 < x < 2