I've been out of a math program for about four years now. My wife is starting a CS degree, and finished her first calculus course last semester.
I tutored calculus throughout my entire undergrad, and have read quite a few calculus and real analysis textbooks. When I'm referring to a candidate for an $x$-value at which a point of extremum is located, I've always called it a critical point.
But my wife's professor taught three different terms:
- Critical number: the $x$-value at which the possible extremum is located
- Critical value: the corresponding $y$-value of the critical number
- Critical point: a pair $(\text{critical number}, \text{critical value})$.
I don't understand the pedagogical value in creating these three terms.
- This doesn't look like standard terminology. I was confused at first when I was trying to help my wife learn about critical points.
- "Critical values" are rarely used in what I specialize in: statistics. I care more about where the extrema are possibly located, rather than what the actual $y$-values are at these possible candidates of extrema. And I like dropping constants whenever possible when it comes to having to find points of extrema, so the $y$-value is useless.
Can someone provide some rationale for why one would want to distinguish between these three bolded terms above?