Disagreement with calculus teacher on a recent test

Question

I had a calculus quiz recently and there was a specific Yes/No question that keeps confusing me and I don't understand my teacher's point.

The question goes: A point on the graph f where f ' is not defined, extreme values can occur on that point. (Yes / No)

I gave examples of functions such as f(x)=x^(1/3) to show that the answer was "No" because in such a function where the derivative is undefined an extreme value is not present (even though a critical point is present),

Is this a flaw in my logic and understanding or is the case I'm coming up with just a deviance from the norm. Would really appreciate if someone could clear my concept and help me with this.

Thanks in advance :)

Answer 1

You exhibited an example that shows that "a point where $f'$ is not defined might not be an extremum". But it doesn't say that it cannot be an extremum.

My example is $f(x)=|x|$. The derivative is undefined at $x=0$, but this point is the (global) minimum. Hence I proved that "a point where $f'$ is not defined can be an extremum".