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resistor circuit [Figure 3.1] in which we wanted to find the maximum power
delivered to a load resistor.) In this section, we will learn the simple Golden
Section rule and the use of the fmin command to solve the simplest forms
of this problem. The important class of problems related to optimizing a
function, while satisfying a number of constraints, will be left to more
advanced courses.
Let us start by reminding ourselves of some terms definitions: The domain
is the set of elements to which a function assigns values. The range is the set
of values thus obtained.

DEFINITION Let I, the domain of the function f(x), contain the point c. We
say that:

1. f(c) is the maximum value of the function on I if f(c) ≥ f(x) for all x ∈ I.
2. f(c) is the minimum value of the function on I if f(c) ≤ f(x) for all x ∈ I.
3. An extremum is the common designation for either the maximum
value or the minimum value.

Using the above definitions, we note that the maximum (minimum) may
appear at an endpoint of the interval I, or possibly in the interior of the
interval:

• If a maximum (minimum) appears at an endpoint, we describe this
extreme point as an endpoint maximum (minimum).
• If a maximum (minimum) appears in the interior of the interval,
we describe this extreme point as a local maximum (minimum).
• The largest (smallest) value among the maximum (minimum) val-
ues (either endpoint or local) is called the global maximum (min-
imum) and is the object of our search.

We note, in passing, the equivalence of finding the local extremum of a func-
tion with finding the zeros of the derivative of this function. The following
methods are suitable when this direct method is not suitable due to a number
of practical complications.
As with finding the zeros of a function, in this instance we will also explore
the graphical method, the simple numerical method, and the MATLAB built-
in commands for finding the extremum.

5.3.1 Graphical Method

In the graphical method, in steps very similar to those described in Section
5.1.1 for finding the zeros of a single variable function, we follow these steps:

© 2001 by CRC Press LLC