# Concavity and Inflection Points: Basic Definitions --- Suppose $f(x)$ is a function. The graph of $f(x)$ is **concave up** at $x = a$ if it is **above** its tangent line on a small interval surrounding $x = a$. ---  ---  --- Suppose $f(x)$ is a function. The graph of $f(x)$ is **concave down** at $x = a$ if the graph of $f(x)$ is **below** its tangent line at $x = a$. ---  ---  --- ## How to Remember? * Any point on a **frown** is **concave down**. * Any point on a **cup** is **concave up**. --- ## Inflection Points An **inflection point** is a place where $f(x)$ changes concavity (from concave up or concave down, to vice versa). It's important to distinguish **inflection points** (where $f(x)$ changes concavity) from **critical points** (where $f'(x) = 0$ or $f'(x)$ DNE). **Example:** Find the critical points and inflection points in this graph of $f(x) = x^3 - 4x$ . <iframe src="https://www.desmos.com/calculator/mzbrz5knyj?embed" width="500" height="500" style="border: 1px solid #ccc" frameborder=0></iframe> **Answer:** The graph of $f(x)$ has critical points at $x \approx -1.155$ and $x \approx 1.155$. The graph of $f(x)$ has an inflection point at $x = 0$. (We'll see how to work this out algebraically in a bit.)  **Another Example:** Here's a video where we discuss how to find the critical and inflection points for several examples of graphs. <iframe width="400" height="215" src="https://www.youtube.com/embed/sWzFUSL8h64" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe> ---