Can a point of inflection be undefined?

Question

Emma Aldridge · Accepted Answer

See the explanation section below.

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]}

Scarlett Allison · Answer

undefined No, a point of inflection cannot be undefined. A point of inflection occurs where the concavity of a curve changes, meaning the second derivative of the function changes sign. However, it's possible for a function to have no points of inflection if its second derivative does not change sign anywhere in its domain.