Can a point of inflection be undefined?

1 Answer
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(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]}