# Can a point of inflection be undefined?

Oct 21, 2015

See the explanation section below.

#### Explanation:

A point of inflection is a point on the graph at which the concavity of the graph changes.

If a function is undefined at some value of $x$, there can be no inflection point.

However, concavity can change as we pass, left to right across an $x$ values for which the function is undefined.

Example

$f \left(x\right) = \frac{1}{x}$ is concave down for $x < 0$ and concave up for $x > 0$.

The concavity changes "at" $x = 0$.

But, since $f \left(0\right)$ is undefined, there is no inflection point for the graph of this function.

graph{1/x [-10.6, 11.9, -5.985, 5.265]}

Example 2

$f \left(x\right) = \sqrt{x}$ is concave up for $x < 0$ and concave down for $x > 0$.

$f ' \left(x\right) = \frac{1}{3 {\sqrt{x}}^{2}}$ and $f ' \left(x\right) = \frac{- 2}{9 {\sqrt{x}}^{5}}$

The second derivative is undefined at $x = 0$.

But, since $f \left(0\right)$ is defined, there is an inflection point for the graph of this function. Namely, $\left(0 , 0\right)$

graph{x^(1/3) [-3.735, 5.034, -2.55, 1.835]}