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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?

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    $\begingroup$ Where are you getting your definition of "singularity"? I had never before heard the idea that the absolute value function has a singularity at zero (it is nondifferentiable there, but singularities are points where the function "blows up"). $\endgroup$ – Xander Henderson Aug 3 '20 at 13:28
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    $\begingroup$ I've heard "stationary point" used to mean what you call a critical point and "critical point" used to mean a point where either the derivative is zero or doesn't exist. $\endgroup$ – J W Aug 3 '20 at 14:03
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    $\begingroup$ @XanderHenderson: The WP article uses |x| as one of its first examples: en.wikipedia.org/wiki/Mathematical_singularity . Singularities are not the same as points where a function blows up. The WP article actually has a pretty nice explanation. $\endgroup$ – Ben Crowell Aug 3 '20 at 19:16
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    $\begingroup$ @BenCrowell Multiple comments in the talk section of that article note that this usage controversial. $\endgroup$ – Xander Henderson Aug 3 '20 at 19:20
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    $\begingroup$ @XanderHenderson: I wouldn't be surprised if it's different in different fields. But for example the notion of a coordinate singularity at the north pole in latitude-longitude coordinates is a totally standard usage of "singularity" in the context of differential geometry when you're talking about coordinate charts on a manifold. $\endgroup$ – Ben Crowell Aug 4 '20 at 22:17
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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.

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    $\begingroup$ It's interesting you use the word "smooth" along with the example $y^3=x$ at the origin :-) and I think it might be worth pointing out to students (those who ask and are motivated, not generic students) that if they think it is unfair this has a critical point there, we can define things to be nice if we swap $x$ and $y$, and that this is actually done in a field that has applications in e.g. robotics. There are some completely scheme-free (even nearly ring-free) books on algebraic curves to send them to if they seem hungry for more. $\endgroup$ – kcrisman Aug 7 '20 at 14:34
  • $\begingroup$ @kcrisman I definitely agree that this example is a great one to play with, and which opens up lots of paths to deeper waters. $\endgroup$ – Steven Gubkin Aug 7 '20 at 17:09

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