Optimization
In this section we are going to look at optimization problems. In optimization
problems we are looking for the largest value or the smallest value that a function can
take. We saw how to solve one kind of optimization problem in the
Absolute
Extrema
section where we found the largest and smallest value that a function would
take on an interval.
In this section we are going to look at another type of optimization problem. Here we
will be looking for the largest or smallest value of a function subject to some kind of
constraint. The constraint will be some condition (that can usually be described by
some equation) that must absolutely, positively be true no matter what our solution is.
On occasion, the constraint will not be easily described by an equation, but in these
problems it will be easy to deal with as we’ll see.
This section is generally one of the more difficult for students taking a Calculus
course. One of the main reasons for this is that a subtle change of wording can
completely change the problem. There is also the problem of identifying the quantity
that we’ll be optimizing and the quantity that is the constraint and writing down
equations for each.
The first step in all of these problems should be to very carefully read the problem.
Once you’ve done that the next step is to identify the quantity to be optimized and the
constraint.
In identifying the constraint remember that the constraint is the quantity that must true
regardless of the solution. In almost every one of the problems we’ll be looking at
here one quantity will be clearly indicated as having a fixed value and so must be the
constraint. Once you’ve got that identified the quantity to be optimized should be
fairly simple to get. It is however easy to confuse the two if you just skim the
problem so make sure you carefully read the problem first!
Let’s start the section off with a simple problem to illustrate the kinds of issues we
will be dealing with here.
Example 1
We need to enclose a field with a fence. We have 500 feet of fencing material and a
building is on one side of the field and so won’t need any fencing. Determine the dimensions of
the field that will enclose the largest area.
Solution
In all of these problems we will have two functions. The first is the function that we are actually

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trying to optimize and the second will be the constraint. Sketching the situation will often help
us to arrive at these equations so let’s do that.
In this problem we want to maximize the area of a field and we know that will use 500 ft of
fencing material. So, the area will be the function we are trying to optimize and the amount of
fencing is the constraint. The two equations for these are,
Okay, we know how to find the largest or smallest value of a function provided it’s only got a
single variable. The area function (as well as the constraint) has two variables in it and so what
we know about finding absolute extrema won’t work. However, if we solve the constraint for one

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