5.5 - ABSOLUTE EXTREMA
I.
Definition: Absolute MAX if f (c)  f (x) for all x in the domain of f.
Absolute MIN if f (c)  f (x) for all x in the domain of f.
Extreme Value Theorem: A function f continuous on a closed interval [a, b] assumes both an absolute
maximum and an absolute minimum on that interval.
 All absolute extrema (if they exist) must always occur at critical values or at endpoints.
1.
2.
3.
4.
f continuous over [a, b]?
Find critical values in (a, b).
Find f(a), f(b), f(c).
Absolute maximum is the largest of step 3. Absolute minimum is smallest of step 3.
Example:
(a)
(b)
Example:
Example:
II. Second Derivative Test for Absolute Extrema: Let f be continuous on I and c the only critical value
in I. Then
Example:
Example:
5.5 HW # 11 – 27 (odd)
III. - OPTIMIZATION - APPLICATIONs OF ABSOLUTE EXTREMA
Word problems MUST be done using CALCULUS! Trial and error methods will earn no credit.
Must include definition of variables, proper equations and a response to the question.
Example: How would you divide a 10-inch line so that the product of the two lengths is a maximum?
5.5 HW # 31 – 39 (odd)