Concavity, Convexity, and Points of Inflection

A graph is said to be concave down (convex up) at a point if the tangent line there lies above the graph in the vicinity of the point; it is concave up (convex down) at a point if the tangent line lies below the graph nearby.

Definition 7.8 (repeated from §7.6.2's numbering — the second derivative version). Let $f(x)$ be a function whose second derivative exists on an open interval $I=(a,b)$. Then $f(x)$ is:

1. concave up on $I$ if $f'(x)$ is strictly increasing on $I$;
2. concave down on $I$ if $f'(x)$ is strictly decreasing on $I$. Analytically, the concavity of a differentiable $y=f(x)$ is captured by testing the second derivative directly: Theorem 7.11 (Test of Concavity).

(i) If $f''(x)>0$ on an open interval $I$, then $f(x)$ is concave up on $I$.

(ii) If $f''(x)<0$ on an open interval $I$, then $f(x)$ is concave down on $I$.

Definition 7.9 (Points of Inflection). The points where the graph of $f(x)$ changes from "concave up to concave down" or "concave down to concave up" are called points of inflection of $f(x)$.

Theorem 7.12 (Test for Points of Inflection).

(i) If $f''(c)$ exists and changes sign as $x$ passes through $c$, then $\big(c,f(c)\big)$ is a point of inflection of the graph of $f$.

(ii) If $f''(c)$ exists at a point of inflection, then $f''(c)=0$.

Locating candidate inflection points. For a "smooth" curve (no sharp corners), points where $f''(x)$ might change sign happen either where $f''(x)=0$ or where $f''(x)$ does not exist.