Question

Can a point of inflection be undefined?

Answer 1

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(x) = 1/x` is concave down for `x < 0` and concave up for `x > 0`.

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

But, since `f(0)` 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(x) = root3x` is concave up for `x < 0` and concave down for `x > 0`.

`f'(x) =1/(3root3x^2)` and `f'(x) =(-2)/(9root3x^5)`

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

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

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