Concavity

The first derivative describes slope; the second derivative describes how that slope is changing.

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.

Second derivative test for concavity

The second derivative is

Where $f''(x)>0$, $f$ is concave up. Where $f''(x)<0$, $f$ is concave down.

This works because positive $f''$ means the first derivative is increasing, while negative $f''$ means it is decreasing.

Candidates for inflection points

Possible inflection inputs occur where

or $f''$ is undefined, provided $f$ is defined there. These are candidates only.

An inflection point requires an actual change in concavity from one side to the other.

Worked example

A zero second derivative without inflection

For $f(x)=x^4$, $f''(x)=12x^2$, which equals zero at $0$ but remains nonnegative on both sides. The graph is concave up on both sides, so $(0,0)$ is not an inflection point.

Again, an equation gives candidates; a sign change gives classification.

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.

Reading graphs of f′ and f″

From a graph of $f'$, the original function is concave up where $f'$ rises and concave down where $f'$ falls. Local extrema of $f'$ can correspond to inflection points of $f$ when the direction of $f'$ changes.

From a graph of $f''$, use whether it lies above or below the axis.

Second derivative and extrema

If $f'(c)=0$ and $f''(c)>0$, then $f$ has a local minimum at $c$. If $f''(c)<0$, it has a local maximum. If $f''(c)=0$, the second derivative test is inconclusive.

This classification rule is related to concavity but does not replace a full interval analysis.

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?

Related topics

• First Derivative Test
• Critical Points
• Curve Sketching
• Second Derivative Test