3.4: Optimization

Many real-world problems involve optimization; that is, finding the global maximum

and minimum values of a function on an interval. For example, a company might be

interested in the maximum amount of profit it can make or in the minimum cost of

producing some product. In this section we will learn how to use limits, derivatives,

and values to find the global extrema of a function on an interval, so that we can

solve such real-world problems.

The point x = c is the location of a global maximum of a function f on an interval

I if f(c) � f(x) for all x 2 I. Sometimes local extrema are also global extrema, and

sometimes they are not. For example, the following four functions each have a local

maximum at the critical point x = 2:

1 2 5

2

4

f(x)

1 2 5

2

4

g(x)

1 2 5

2

4

h(x)

1 2 5

2

4

k(x)

To find a global extremum in an interval I, we need to check:

• the function value at critical points inside I,

• the function values at any closed endpoints of I,

• the limits at any open endpoints or non-domain points of I.

1

3.4: Optimization 2

1. Translating Word Problems into Mathematical Problems

Finding the global extrema of a function f on an interval I is a mathematical problem

that we can tackle with function values, limits, and derivatives. Real-world problems

are less straightforward, since they are expressed in full sentences in which the vari-

ables, constants, functions, and relationships are described in words instead of in

mathematical notation. To solve a real-world optimization problem we must first

translate it into a mathematical global extrema problem. Once we solve the mathe-

matical problem, we can translate the solution back into the real-world context.

Example 1.1. Calvin throws a baseball straight into the air at 50 feet per second,

releasing the ball when his hand is five feet above the ground. How high did Calvin

throw the ball? How long was the ball in the air for?

The position s(t) of an object thrown upward is: s(t) = �16t2 + v0 t+ s0, where v0 is

the initial upward velocity and s0 is the initial position.

3.4: Optimization 3

Example 1.2. Find the global maximum and minimum values, if any, of the function

f(x) = 2x3 � 15x2+ 24x+ 20 on the interval (0, 6) .

3.4: Optimization 4

Example 1.3. Farmer Joe wants to build a rectangular chicken pen for his chickens.

He wants to build the pen so that it has the largest area possible, and he only has 100

feet of chicken-wire fencing. What dimensions should he use for his pen?

3.4: Optimization 5

Example 1.4. A rectangular storage container with an open top is to have a volume

of 10 m3.The length of its base is twice the width. Material for the base costs $10

per square meter. Material for the sides costs $6 per square meter. Find the cost of

materials for the cheapest such container.

3.4: Optimization 6

Example 1.5. A Norman window has the shape of a rectangle surmounted by a

semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle.

See the figure below.) If the perimeter of the window is 32 ft, find the value of x so

that the greatest possible amount of light is admitted.

3.4: Optimization 7

2. Section Recap

What are some take-away concepts from this section?