Math 140 Section 4.1

Maximum and Minimum Values

Definitions, Theorems and Other Useful Information

1. Defn: A function f has an absolute maximum (or global maximum) at c if in the domain of f. The number is called the maximum value of f on its domain. Similarly, f has an absolute minimum at c if in the domain of f and the number is called the minimum value of f on its domain. The maximum and minimum values of f are called the extreme values of f.

2. Defn: A function f has a local maximum (or relative maximum) at c if when x is near c (i.e. in some open interval containing c). Similarly, f has a local minimum at c if when x is near c.

3. The Extreme Value Theorem: If f is continuous on a closed interval [a,b], then f attains an absolute maximum value and an absolute minimum value at some numbers c and d in [a,b].

4. Fermat’s Theorem: If f has a local maximum or minimum at c, and if exists then (Note: The converse is not necessarily true.)

5. Defn: A critical number of a function f is a number c in the domain of f such that either or does not exist.

6. The Closed Interval Method: To find the absolute maximum and minimum values of a continuous function f on a closed interval [a,b]:

(i) Find the values of f at the critical numbers of f in (a,b).

(ii) Find the values of f at the endpoints of the interval.

(iii) The largest of the values from Steps (i) and (ii) is the absolute maximum value; the smallest of these values is the absolute minimum value.