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In some texts, a critical point is when the derivative exists and is zero, and a singular point is when the derivative does not exist. So I suppose, at $x=0$, $|x|$ would have a singular point while $x^2$ would have a critical point.
I've usually seen critical points include both cases, but I'm wondering if there's an advantage to using the more precise exclusive definition?
The main reason we have the word "critical point" is for the first derivative test. The statement is slightly easier if you only have to say "critical point" instead of "critical point or singular point". However, students might actually remember to check for singular points if they had a different word for this case. So I can see both sides of the argument.
An activity I do like to do is present several pictures, and get students to capture the behavior symbolically. I draw pictures of a function which looks locally like $x \mapsto x^2$, $x \mapsto x^3$, $x \mapsto |x|$, $x \mapsto \sqrt[3]{x}$, $x \mapsto |\sqrt[3]{x}|$, $x \mapsto x\sin(1/x)$, $x \mapsto x^2\sin(1/x)$ (the point of interest for all of these functions is the origin, but the sketch I draw could be centered anywhere). Students think of some of these graphs as smooth even when the function isn't differentiable (like $x \mapsto \sqrt[3]{x}$). So we spend time just playing with these, writing limits of SDQs which represent the different situations, and play with "classifying" these different kinds of situations. This is good preparation for understanding the diversity of different critical points which are possible.
