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Maximum and Minimum Values
Maximum and Minimum V
UAE_Math_Grade_12_Vol_1_SE_718383_ch3 GO01962-Smith-v1.cls July 4, 2016 13:38 alues

y To remain competitive, businesses must regularly evaluate how to minimize waste and
Has no maximize the return on their investment. In this section, we consider the problem of
absolute finding maximum and minimum values of functions. In section 4.7, we examine how
maximum to apply these notions to problems of an applied nature.
We begin by giving careful mathematical definitions of some familiar terms.

c
x
DEFINITION 3.1
For a function f defined on a set S of real numbers and a number c ∈ S,
f (c)
Absolute (i) f(c) is the absolute maximum of f on S if f(c) ≥ f(x) for all x ∈ S and
minimum (ii) f(c) is the absolute minimum of f on S if f(c) ≤ f(x) for all x ∈ S.

FIGURE 4.24a An absolute maximum or an absolute minimum is referred to as an absolute
extremum. (The plural form of extremum is extrema.)
y

Absolute The first question you might ask is whether every function has an absolute maxi-
maximum mum and an absolute minimum. The answer is no, as we can see from Figures 4.24a
f (c) and 4.24b.

x EXAMPLE 3.1 Absolute Maximum and Minimum Values
c
2
(a) Locate any absolute extrema of f(x) = x − 9 on the interval (−∞, ∞). (b) Locate
2
Has no any absolute extrema of f(x) = x − 9 on the interval (−3, 3). (c) Locate any absolute
2
absolute extrema of f(x) = x − 9 on the interval [−3, 3].
minimum
Solution (a) In Figure 4.25, notice that f has an absolute minimum value of f(0) =
FIGURE 4.24b −9, but has no absolute maximum value.
(b) In Figure 4.26a, we see that f has an absolute minimum value of f(0) =−9.
Your initial reaction might be to say that f has an absolute maximum of 0, but f(x) ≠ 0
y No absolute
maximum for any x ∈ (−3, 3), since this is an open interval. Hence, f has no absolute maximum
on the interval (−3, 3).
y y
No absolute Absolute maximum
maximum f(-3) = f(3) = 0
x
-3 3 -3 3 -3 3
x x

f(0) = -9
(Absolute minimum) f(0) = -9 f(0) = -9
(Absolute minimum) (Absolute minimum)
FIGURE 4.25
2
y = x − 9 on (−∞, ∞) FIGURE 4.26a FIGURE 4.26b
Copyright © McGraw-Hill Education
2
2
y = x − 9on(−3, 3) y = x − 9on[−3, 3]
(c) In this case, the endpoints 3 and −3 are in the interval [−3, 3]. Here, f assumes
its absolute maximum at two points: f(3) = f(−3) = 0. (See Figure 4.26b.)

250
250 | Lesson 4-3 | Maximum and Minimum Values