Why are critical points called critical? For a function $y = f(x)$, a number $x_0$ is called $\textit{critical}$ if either $f'(x_0) = 0$ or $f'(x)$ does not exist. Sometimes the term $\textit{stationary}$ is used, but it is by far less popular. My question is

Why is the word "critical" used in this case as terminology? What makes $x_0$ critical if $f'(x_0) = 0$? 

Of course the tangent line being horizontal or not being able to draw a tangent line gives local minimums and maximums. So why are maximums and minimums "critical"? It seems that "stationary" is more appropriate. So I am puzzled as to why the latter is less popular.

Critical as in "important" or "key" (for analyzing the behavior of the function).

For a continuous function from $\mathbb{R}$ to $\mathbb{R}$, the critical points may or may not correspond to actual turning points, but they are the only places where a turning point is possible. Thus, one could say that analyzing the local behavior of the function at or near such points is critical to understanding the behavior of the function. 

But note: The term "critical point" is not synonymous with the term "stationary point". All stationary points are critical, but not every critical point is stationary.

If $f\colon\mathbb{R}\to\mathbb{R}$ is continuous,


*

*$x=c\;$ is a stationary point for $f$ if $f'(c)=0$.

*$x=c\;$ is a critical point for $f$ if either $f'(c) = 0\;$ or $f'(c)$ does not exist.

