Concavity

Concave up and concave down

A graph is concave up on an interval when its tangent slopes increase as $x$ increases. It is concave down when tangent slopes decrease.

Concave up resembles a cup and concave down resembles a cap, but slope behaviour is the reliable definition—even when the graph itself is below an axis.

Worked example

Relationship to increasing and decreasing

Concavity and direction are independent:

• increasing and concave up: positive slopes becoming larger;
• increasing and concave down: positive slopes becoming smaller;
• decreasing and concave up: negative slopes becoming less negative;
• decreasing and concave down: negative slopes becoming more negative.

Separating the sign of $f'$ from the sign of $f''$ prevents visual misconceptions.

Modelling interpretation

If position is $s(t)$, then $s''(t)$ is acceleration. Positive acceleration means velocity is increasing, not necessarily that the object moves forward. A moving object can have negative velocity and positive acceleration while slowing in the negative direction.

In economics or growth models, concavity can signal increasing or diminishing marginal change.

Common mistakes

Using whether $f$ is positive to decide concavity. Use change in slope or sign of $f''$.

Calling every solution of $f''=0$ an inflection point. Concavity must change.

Equating concave up with increasing. A concave-up graph may still decrease.

Ignoring undefined second-derivative candidates. Check them if $f$ exists there.

Using $f''=0$ to claim no information anywhere. Test intervals around the candidate.

Quick self-check

• What is $f''(x)$ and where can its sign change?
• Are all candidates in the domain of $f$?
• What sign does $f''$ have on each interval?
• Does concavity actually change at each proposed inflection point?
• Are increasing/decreasing and concavity being described separately?
• Does the interpretation match the units and context?