Applied Maximum and Minimum Problems
Read this section to learn how to apply previously learned principles to maximum and minimum problems. Work through practice problems 1-3. There is no review for this section; instead, make sure to study the problems carefully to become familiar with applied maximum and minimum problems.
Practice Answers
Practice 1:
which is
defined for all so the only critical numbers are the endpoints and and the places where equals , at and (but is not in the interval so it is not practical for this applied
problem).
The maximum volume must occur when , or ):
Fig. 31
Practice 2: (a) We have feet of fencing. (See Fig. 32). Our assignment is to maximize the area of the garden: (two variables). Fortunately we have the constraint that so , and our assignment reduces to maximizing a function of one variable:
so is concave down, and has a maximum at ,
The maximum area is square feet when . and . The maximum area garden is a square.
Fig. 32
(b) This is very similar to part (a) except we have feet of fencing instead of feet. so , and we want to maximize so when and . The maximum area is square feet and that occurs when the garden is a square and half of the new fence is used on each of the two new sides.
Practice 3: Cost (area of top) (area of sides) (area of bottom)
so our assignment is to minimize , a function of two variables and .
Fortunately we also have the constraint that volume so . Then so if so and in. Then . (. for all so is concave up and we have found a minimum of .)