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Sec. 4.1: Maximum and Minimum Values Calculus provides e§ective, power means for solving a host of optimization problems, such as maximizing the proÖt of a company or minimizing the pollution of an industrial process
Maximum and Minimum Values
M is the (global) maximum of a function f on a set D if
f (c) = M for some c in D and f (x) M for all x in D:
m is the (global) minimum of a function f on a set D if
f (d) = m for some d in D and f (x) m for all x in D:
Closely related (but di§erent) concepts follow:
M is a local maximum of a function f if
f (c) = M for some c and f (x) M for all x near c:
m is a local minimum of a function f on a set D if
f (d) = m for some d and f (x) m for all x near d:
The statement ìfor all x near cîmeans there is an open interval I that contains c such that f (x) M for all x in I. Consequently, if the domain of f (x) is the closed interval [a; b] ; then neither f (a) nor f (b) is a local max or min; however, either may turn out to be the global max or min. This subtle distinction is important for what follows. The following observation also is important: If the domain of f (x) is the closed interval [a; b] and if either the global max or min occurs at a point inside [a; b], the the global max or min is also a local max or min.
Max-Min Theorem: If f (x) is continuous on a closed interval [a; b] ; then f takes on its maximum value M and its minimum value m at certain points in [a; b] ; that is, there are points c and d in the interval [a; b] such that
m = f (d) f (x) f (c) = M
for all x in [a; b] :
Note Well: If either of the words ìcontinuousî or ìclosedî is omitted from the statement of the Max-Min Theorem, then the statement is false. The max-min theorem tell us that maximum and minimum values are waiting to be found, at least for a function f (x) continuous on a closed interval [a; b] ; but the theorem does not tell you how to Önd the maximum and minimum values. That is where calculus come into play.
On Locating Extreme Values (AKA local or global maximum or mini- mum values)
Theorem 1: If f (x) has a local max or min at c; then either
f 0 (c) = 0
or
f 0 (c) does not exist.
DeÖnition: Let f (x) be deÖned for all x near c: The c is called a critical point (number) of f if either f 0 (c) = 0 or f 0 (c) does not exist.
With this language, Theorem 1 can be expressed as
Theorem 1: If f (x) has a local max or min at c; then c is a critical point of f:
The importance of this result for us is that we can limit the search for points where f may have a local max or min to the critical points of f: Because of an earlier remark (which one?), Önding critical points also helps us Önd global extreme values. Indeed, the foregoing leads to the following procedure for Önding global extreme values:
The Closed Interval Max-Min Method: Let f (x) be continuous on a closed interval [a; b] : Step 1. Find f (a) and f (b) : Step 2. Find f (c) for all critical points c of f in (a; b) : Then the global max (respectively, min) of f is the largest (respectively, small- est) of the numbers found in Steps 1 and 2.
Why or when is this method likely to be helpful?
