5.6 Determining Concavity of Functions over Their Domains

Concavity and the second derivative

A function is concave up on an interval where f''(x) > 0, meaning the slope (f') is increasing. A function is concave down where f''(x) < 0, meaning f' is decreasing. To find concavity, take the second derivative, find where it's zero or undefined, and test the sign of f'' in each resulting interval.

Relating f, f', and f''

Concavity describes the graph of f, but it's determined by f'', which is the derivative of f'. If f' is increasing, f is concave up (graph curves like a cup); if f' is decreasing, f is concave down (graph curves like a frown). This links directly to whether f' itself is increasing or decreasing.

Points of inflection

An inflection point occurs where concavity changes — f'' switches sign (from + to − or − to +). At such a point, f''(x) = 0 or is undefined, AND the sign of f'' must actually change on either side. Just having f''(x) = 0 is not enough — always verify the sign change before calling it an inflection point.

Key terms

Concave up condition

f''(x) > 0 on an interval; f' is increasing there.

Concave down condition

f''(x) < 0 on an interval; f' is decreasing there.

Inflection point

A point where the concavity of f changes, i.e., f'' changes sign (and f''(x) = 0 or is undefined there).

Necessary but not sufficient condition for inflection point

f''(x) = 0 or undefined at that point — must also confirm the sign of f'' actually changes.

How to test concavity

Find f'', locate where it's zero/undefined, then check the sign of f'' in each interval between those points.

Try an AP-style question

Where is f(x) = x³ − 3x² concave up?

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