A function has an global maximum at a if f(a) ≥ f(x) for all x in D, domain of f. f(a) is called the maximum value of f on D.
A function has an global minimum at a if f(a) ≤ f(x) for all x in D, domain of f. f(a) is called the minimum value of f on D.
If a function is a continuous on a closed interval [a, b], then the function attains both an absolute maximum value M and an absolute minimum value m in [a, b]. It means, there are numbers c and d in [a, b] with f(c) = M and f(d) = m and m ≤ f(x) ≤ M for every other x in [a, b].
This is called Extreme Value Theorem. The theorem doesn't apply to functions which are not continuous on [a, b].
A function has a local maximum at a point a if f(a) ≥ f(x) for all x in some open interval containing a.
A function f has a local minimum at a point a if f(a) ≤ f(x) for all x in some open interval containing a.
If f has a local maximum or minimum at a, and if f'(a) exists, then f'(c) = 0.