Concavity and Inflection Points of a Function

Concavity of a Function

Before we describe what an inflection point is, it is first important to describe what it means for a curve to be "concave up" or "concave down".

Definition: A function $f$ is described to be Concave Up on the interval $(a, b)$ if the slopes of the tangent lines from $a$ to $b$ are increasing, that is if $x, y \in (a, b)$ where $x < y$, then $f'(x) < f'(y)$. Similarly, $f$ is described to be Concave Down on the interval $(a, b)$ if the slopes of the tangent lines from $a$ to $b$ are decreasing, that is if $x, y \in (a, b)$ where $x < y$, then $f'(x) > f'(y)$.

For example, consider the curve $f(x) = x^3 + 1$ shown below:

We can clearly see that on the interval $(-\infty, 0)$, the function is concave down, while on the interval $(0, \infty)$, the function is concave up. An easy way to remember concavity is by thinking that "concave up" is a part of a graph that looks like a smile, while "concave down" is a part of a graph that looks like a frown.

Inflection Points of Functions

Definition: If $f$ is a continuous function at the point $P(a, f(a))$ is said to be an Inflection Point of $f$ if at the point $P$, $f$ changes from concave up to concave down OR from concave down to concave up.