MATH 241, LECTURE 21
1. Applied optimization problems
First, we should remember how to find the largest and/or smallest values of a function over some
interval.
ā¢Find all stationary points (by finding the derivative and setting it to zero).
ā¢Find all singular points.
ā¢Find the endpoints (these are often given).
ā¢Compared values at all of these points - the largest is the max and the smallest is the min.
In applied problems one usually has to work in order to find a single function to maximize. One needs
to keep track of relationships and constraints.
Example 1. Suppose advertising costs $1000 per unit (say for magazine adds), and product development
costs $20000 per unit. Suppose that the profits generated from xunits of advertising and yunits of product
development are xy2thousands of dollars. If a company has $10000 to spend on advertising and product
development together, how should the money be allocated in order to maximize profits?
Applied problems are not as ācut and driedā as other exercises, but there are some elements common
to most such problems.
ā¢Identify all variables involved. They often need to be named.
ā¢Identify which variable needs to be maximized or minimized.
ā¢Identify all relationships between and constraints on variables.
ā¢(Optional) Restate problem abstractly in the form āfind the maximum of the function ... with the
constraint(s) that ...ā
ā¢Use the relationships to express one variable as a function of one other variable.
ā¢Use our optimization techniques above to find the minimum or maximum of the appropriate
variable.
Example 2. Farmer Fred has 100ft of fencing to use to enclose his sheep in a rectangular area next to a
river. What is the largest area which he can enclose?
Example 3. Suppose the top and bottom of a box is made of a metal which costs 10 cents per square
centimeter and the sides are made of a metal which costs 12 cents per square centimeter. What is the
largest volume can which can be made from two dollars of material?
Example 4. Suppose the daily production level at a factory is modeled by a Cobb-Douglas production
function P=L0.7C0.3, where Lis the number of workers and Cis the cost of materials measured in
thousands of dollars. If each worker costs the company $200 per day, what is the maximum production
level which can be achieved with a total cost of $10000 per day.
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