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.

According to Earliest Known Uses of Some of the Words of Mathematics:

CRITICAL POINT is found in 1871 in A General Geometry and Calculus by
  Edward Olney [University of Michigan Historic Math Collection].

I'm somewhat surprised it is that recent, but I guess the systematic consideration of the possibility of non-differentiability didn't really happen before the mid-19'th century.
EDIT:
As it turns out, Olney's book is available on-line and searchable.  The first appearance of "critical" 
is on p. 118:

As $y = f(x)$, this problem is the ordinary one of maxima and minima of functions of a single variable, treated in the Calculus.  Hence we find the values of $x$ which render $\dfrac{dy}{dx} = 0$ or $\infty$, as critical values, i.e., values to be examined, and at which the property exists, if it exist at all.

