Is $f'(x)=0$ true at a cusp? If I am given a graph that has a cusp and I am asked to find every point $x$ where $f'(x)=0$ is satisfied, does this include the cusp? I know that when the derivative of a function equals zero this means that there is a horizontal tangent at that point, I also know that the derivative does not exist at a cusp. However, a cusp does have a horizontal tangent but is not differentiable at that point, so do we include the point $x$ where the cusp is when the question is asking us where $f'(x)=0$? By cusp I mean when $$\lim_{x \to a^{+}} f'(x)=+ \infty \text{ and } \lim_{x \to a^{-}} f'(x)=- \infty$$

if the derivative does not exist it is, in particular _not_ $=0$