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4-59 QC: OSO/OVY T1: OSO October 25, 2018 17:24 SECTION 4.7 • • Optimization 269
determined by intersections of the graphs of y = r(1 − x∕k) 2. Suppose that f is a function with two derivatives and
′′
′
and y = x . Here, x = N∕A, r is proportional to the that f(a) = f (a) = 0 but f (a) ≠ 0 for some number a.
1 + x 2 Show that f(x) has a local extremum at x = a. Next, sup-
birthrateofthebudwormsandkisdeterminedbytheamount
of food available to the budworms. Note that y = r(1 − x∕k) pose that f is a function with three derivatives and that
′
′′
′′′
is a line with y-intercept r and x-intercept k. How many so- f(a) = f (a) = f (a) = 0 but f (a) ≠ 0 for some number a.
lutions are there to the equation r(1 − x∕k) = x ? (Hint: Show that f(x) does not have a local extremum at x = a.
(k)
1 + x 2 Generalize your work to the case where f (a) = 0 for
The answer depends on the values of r and k.) One current k = 0, 1, ... , n − 1, but f (a) ≠ 0, keeping in mind that
(n)
theory is that outbreaks are caused in situations where there there are different conclusions depending on whether n is
are three solutions and the population of budworms jumps odd or even. Use this result to determine whether f(x) =
from a small population to a large population. x sin x or g(x) = x sin(x ) has a local extremum at x = 0.
2
2
2
4.7 OPTIMIZATION

Everywhere in business and industry today, we see people struggling to minimize waste
and maximize productivity. In this section, we bring the power of the calculus to bear
on a number of applied problems involving finding a maximum or a minimum. We
start by giving a few general guidelines.

• If there’s a picture to draw, draw it! Don’t try to visualize how things look in your
head. Put a picture down on paper and label it.
• Determine what the variables are and how they are related.
• Decide what quantity needs to be maximized or minimized.
• Write an expression for the quantity to be maximized or minimized in terms of
only one variable. To do this, you may need to solve for any other variables in
terms of this one variable.
• Determine the minimum and maximum allowable values (if any) of the variable
you’re using.
• Solve the problem and be sure to answer the question that is asked.
We begin with a simple example where the goal is to accomplish what businesses
face every day: getting the most from limited resources.

OR
EXAMPLE 7.1 Constructing a Rectangular Garden
of Maximum Area
You have 40 (linear) feet of fencing with which to enclose a rectangular space for a
garden. Find the largest area that can be enclosed with this much fencing and the
dimensions of the corresponding garden.

Solution First, note that there are lots of possibilities. We could enclose a plot that
is very long but narrow, or one that is very wide but not very long. (See Figure
FIGURE 4.78 4.78.) We first draw a picture and label the length and width x and y, respectively.
Possible plots
(See Figure 4.79.)
We want to maximize the area,
A = xy.

y However, this function has two variables and so, cannot be dealt with via the
means we have available. Notice that if we want the maximum area, then all of the
fencing must be used. This says that the perimeter of the resulting fence must be
′
40 and hence,
x
40 = perimeter = 2x + 2y. (7.1) Copyright © McGraw-Hill Education
FIGURE 4.79
Rectangular plot Notice that we can use (7.1) to solve for one variable (either one) in terms of the other.

288 | Lesson 4-6 | Overview of Curve Sketching