Calculus: Analytic Geometry and Calculus, with Vectors

5.2 Trends, maxima, and minima 295

Definition 5.24 Let f be defined over a nonempty set S in El. We say

that f has a local (or relative) maximum over S at xo and that f(xo) is a local

maximum off over S, if there is a positive number h such that f(x) 5 f(xo)

whenever x is in S and Ix - xol < A. We say that f has a global maximum

(or absolute maximum) over S at xo if f(x) < f(xo) whenever x is in S. A

local minimum and a global minimum are similarly defined, the relation

f (x) < f(xo) being replaced by f (x) > f(xo).

Applications of these definitions can be quite diverse. For example,

f(x) might be the number of telephones ringing in Chicago x hours after

the beginning of the nineteenth century and S might be the whole set of

positive integers or any other set of numbers we wish to select. For

many purposes it suffices to see how these definitions are applied when

S is an interval and f is differentiable over the interval. Let f be the

function whose graph is shown in Figure 5.25. Assuming that there is

nothing deceptive about the graph, we can see that f is increasing over
the intervals a < x < xl and x2 < x S x4 and that f is decreasing over the
intervals xl < x -< X2 and x4 < x < b. Supposing that f'(x3) = 0 so
that the graph has a horizontal tangent at the point (x3, f(x3)), it appears
that f has local maxima at xl and x4, a global maximum at x4, local
minima at a, x2, and b, and a global minimum at a. There is neither a
local minimum nor a local maximum at x3 even though f'(x3) = 0. We
have described the trends (the increasings and the decreasings) and the
extrema (the maxima and minima) of f.
Sometimes we are required to obtain information about a function f
when we do not have a graph of f but do have a formula which determines
values of f(x) for different numbers x. As the discussion of Figure 5.25
indicates, it is often quite impossible to give precise information about a
differentiable function f until we have found the values of x for which
f'(x) = 0. These are the values of x for which the graph of f has hori-
zontal tangents, and they are called critical values of x. After we succeed
in finding when f'(x) = 0, when f'(x) > 0, and when f'(x) < 0, we may
find it convenient to construct a figure
more or less like Figure 5.251 in which we Figure 5.251
(i) mark the points at which f' (x) = 0 and +++ _ ++++ ......
the graph of f has horizontal tangents, (ii) a xl x2 x3 x4 b