Theorem 4.2, the Extreme-Value Theorem:
Every continuous function f defined on a closed interval [a, b] will attain both a global minimum and a global maximum on [a, b].
Definition: Critical Point:
For a function f, a critical point x = c is a point where either
-     f ¢ (c) is undefined, or
-     f ¢ (c) = 0.
Theorem 4.1:
If f is defined on (a, b) and has a local extremum at x = c in (a, b), then if f ¢(c) exists,  f ¢(c) = 0.
Local extrema of continuous functions occur at critical points or endpoints of an interval.
The same must be true for global extrema.