Can a point of inflection be undefined?

1 Answer
Oct 21, 2015

Answer:

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]}