If the first derivative has a cusp at x=3, is there a point of inflection at x=3 even though the second derivative doesn't exist there?

1 Answer
Jim H
Apr 11, 2018

It depends, in part, on the definition of inflection point being used.

Explanation:

I have seen some who insist that the second derivative must exist to have an IP.

I am more used to the definition:

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

So I consider the point #(0,0)# an inflection point for #f(x) = root(3)x# in spite of the non-existence of #f'(0)# and #f''(0)#.

Similarly, the function #f(x) = 1/2xabsx# has derivative #f'(x)=absx#. So the derivative has a cusp at #0#.
Since the graph of #f# is concave down on #(-oo,0)# and concave up on #(0,oo)# and #f(0)# exists (it is # = 0#), I count #(0,0)# as an inflection point.

In the graph below, you see #f# in blue, #f'# in red and #f''# in orange.

enter image source here

Translate 3 to the right to get an example at #3#.

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