171
C H A P T E R 5
Optimization
In this chapter we will start on optimization, the process of finding the largest or smallest value
a function can take on as well as values that are larger (or smaller) than all nearby values. is
chapter only lets us optimize functions of one independent variable, but it sets the stage for more
complex types of optimization later in the series.
5.1 OPTIMIZATION WITH DERIVATIVES
As is often the case, our first step is to define our terms. We need terminology to sort out
different types of maximum and minimum value.
Definition 5.1 If a function f .x/ is defined on an interval or collection of intervals I, then the
global maximum of f .x/ is the largest value f .x/ takes on anywhere on I.
Definition 5.2 If a function f .x/ is defined on an interval or collection of intervals I, then the
global minimum of f .x/ is the smallest value f .x/ takes on anywhere on I.
Definition 5.3 If a function f .x/ is defined near and including x D c and, in some interval
I D .c ı; c C ı/, we have that, for all a 2 I , f .c/ f .a/, then we say f .x/ has a local
maximum at x D c.
Definition 5.4 If a function f .x/ is defined near and including x D c and, in some interval
I D .c ı; c C ı/, we have that, for all a 2 I , f .c/ f .a/, then we say f .x/ has a local
minimum
at
x
D
c
.
We use the term optima for values that are maxima or minima. Figure 5.1 demonstrates the
different sorts of optima.
e key to finding optima is the behavior of the derivative. Notice that, at an optimum, the
function changes between increasing and decreasing. is means that optima, at least ones that
don’t occur at the beginning and end of an interval, are a kind of point we’ve seen before: they
172 5. OPTIMIZATION
f .x/
5
-4
-5 5
global minimum
local maximum
local minimum
global maximum
Figure 5.1: is picture shows examples of local and global optima.
are critical points.
Knowledge Box 5.1
Critical points in optimization
Except at the boundaries of an optimization problem, the optima of a
continuous, differentiable function occur at critical points.
If we consider the tangent line at a critical point, we get a geometric description of the behavior
of the derivative. A line with a slope of 0 is a horizontal tangent line. is tells us that optima
can occur at horizontal tangent lines.
A function that diverges to infinity as jxj gets large may fail to have a global maximum or
minimum. Similarly, vertical asymptotes inside the area where we are optimizing can cause
trouble. Having said that, the other place besides critical points where optima can occur is at the
boundaries of the region where we are optimizing. In general, when optimizing a continuous,
differentiable function, we check the critical points and the points at the boundaries of the area
we are optimizing.
5.1. OPTIMIZATION WITH DERIVATIVES 173
e next example asks you to optimize a quadratic equation—something that should be
easy—but it asks you to do it on a bounded domain. A quadratic that opens upward has a
unique minimum. As well as the critical points, we must also check the x values at its endpoints.
Example 5.1 Find the global maximum and minimum of g.x/ D x
2
4 on the interval [-2,3 ].
Solution:
e critical point of this function is easy: g
0
.x/ D 2x. So x D 0 is the critical value, and there
is a critical point at (0,-4). e boundaries of the optimization area are x D 2 and x D 3.
Since g.2/ D 0 and g.3/ D 5, we have three candidate points: (-2,0), (0,-4), and (3,5). e
smallest y-value is 4 making (0,-4) the point where the global minimum occurs, while 5 is the
largest y-value making (3,5) the point where the global maximum occurs. Let’s look at a picture.
5
-5
-3 3
global minimum
global maximum
horizontal tangent line.
e function g.x/ D x
2
4 on the interval [-2,3]
˙
Knowledge Box 5.2 summarizes what we have so far.
174 5. OPTIMIZATION
Knowledge Box 5.2
e Extreme Value eorem
e global maximum and minimum of a continuous, differentiable func-
tion must occur at critical points (horizontal tangents) or at the bound-
aries of the domain where optimization is taking place.
An important point is that critical points don’t have to be maxima or minima. Let’s look at an
example of this. Consider the function
h.x/ D
x
3
9
;
shown in Figure 5.2.
4
-4
-3 3
Figure 5.2: e function h.x/ D
x
3
9
.
Since f
0
.x/ D
1
3
x
2
, it’s easy to see there is a critical point at .0; 0/—but it’s not a maximum
or a minimum. is then opens the question, how do we tell if a critical point is a maximum,
a minimum, or neither? We have already developed one useful tool: the sign chart for the first
derivative. Let’s look at the sign chart for h.x/ D
x
3
9
:
.1/ C C C .0/ C C C .1/
5.1. OPTIMIZATION WITH DERIVATIVES 175
e function increases to zero and then continues to increase. ere are four ways a sign chart
can shake out which together are the first derivative test for the nature of an optimum.
Knowledge Box 5.3
e first derivative test for optima
f
0
.x/ chart near p Conclusion:
C C C.p/ C CC no optimum at p
C C C.p/ maximum at p
.p/ C CC minimum at p
.p/ no optimum at p
Another method of spotting a potential maximum or minimum is to look at the second
derivative. A function that is curved downward at a critical point has a maximum at that critical
point; a function that is curved upward at a critical point has a minimum at that critical point.
is technique is called the second derivative test.
Knowledge Box 5.4
e second derivative test for optima
f
00
.x/ > 0 f
00
.x/ < 0
Concave up Concave down
Minimum Maximum
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