# What is different between critical point and inflection point?

Jul 10, 2015

There seem to be two definitions of "critical point" in use. But with that in mind:

#### Explanation:

A critical number for function $f$, is a number in the domain of $f$ where $f ' \left(x\right) = 0$ or $f ' \left(x\right)$ does not exist.
These are the numbers at which Fermat's Theorem on local extrema tells us $f$ could have a local maximum or local minimum value.

In some usages "critical point" is synonymous with this "critical number".
In other usages a critical point is a point $\left(x , f \left(x\right)\right)$ with $f ' \left(x\right) = 0$ or $f ' \left(x\right)$ does not exist.

An inflection point for the graph of function $f$ is a point on the graph at which the concavity of the graph changes.
For twice differentiable functions, this is a point on the graph of $f$ at which $f ' ' \left(x\right)$ changes sign.

Students sometimes use "inflection point" to mean an $x$ value at which the concavity changes.

The difference is illustrated by $f \left(x\right) = \frac{1}{x}$ which is concave down on $\left(- \infty , 0\right)$ and concave up on $\left(0 , \infty\right)$.
In the "point" usage, the graph has no inflection point, because there is no point on the graph where concavity changes.