Concavity and Points of Inflection

Two cars can both be speeding up the road, one gaining speed faster and faster, the other gaining speed more and more slowly. Both position graphs go up, but they bend differently. That bend is concavity. You met it with tangent line approximations; now it becomes a tool for describing a whole graph, and the second derivative is how you measure it.

Concave up and concave down

On an interval, the graph of $f$ is concave up if $f'$ is increasing there: the slopes get larger as you move right, and the graph bends upward like a cup. It is concave down if $f'$ is decreasing: the slopes get smaller, and the graph bends downward like a cap.

Since $f''$ is the derivative of $f'$, its sign tells whether $f'$ increases:

• $f''(x) > 0$ on an interval: $f'$ increasing, graph concave up.
• $f''(x) < 0$ on an interval: $f'$ decreasing, graph concave down.

Concavity is independent of direction. A graph can be increasing and concave down (rising, but leveling off), or decreasing and concave up (falling, but leveling off).

The tangent lines show it too. On a concave-up stretch the graph lies above its tangent lines, so a tangent line approximation there underestimates; on a concave-down stretch it overestimates.

Points of inflection

A point of inflection is a point on the graph where the concavity changes, from up to down or from down to up. For a function with a second derivative, that means $f''$ changes sign there. Equivalently, $f'$ changes from increasing to decreasing, or from decreasing to increasing, so $f'$ has a relative maximum or minimum at that $x$.

To find them:

1. Find $f''(x)$.
2. Find where $f''(x) = 0$ or $f''(x)$ does not exist. These are only candidates.
3. Check the sign of $f''$ on each side of each candidate. Keep the ones where the sign changes (and where $f$ is defined).

A zero of $f''$ is not enough

For $f(x) = x^4$, $f''(x) = 12x^2$, so $f''(0) = 0$. But $f'' > 0$ on both sides of 0: the graph is concave up the whole way, and there is no point of inflection. This is the same trap as "$f'(c) = 0$ means an extremum": a zero is a candidate, and the sign change is what decides. A justification that says only "$f''(c) = 0$" earns nothing.

Concavity from a graph of $f'$

This is the most tested version. Given the graph of $f'$:

The common error is to put inflection points at the zeros of $f'$. Zeros of $f'$ are where $f$ may have extrema; the turns of $f'$ are where $f$ inflects.

Concavity in context

If $N(t)$ counts people who have heard a rumor, $N'(t)$ is how fast it spreads. If $N'$ increases until $t = 3$ and decreases after, the spread is fastest at $t = 3$: the graph of $N$ has a point of inflection there, changing from concave up to concave down. In words, "increasing at a decreasing rate" means $f' > 0$ and $f'' < 0$.