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We have seen that a function may or may not have absolute extrema. In example
3.1, the function failed to have an absolute maximum, except on the closed, bounded
interval [−3, 3]. Example 3.2 provides another piece of the puzzle.
EXAMPLE 3.2 A Function with No Absolute Maximum or Minimum
y
Locate any absolute extrema of f(x) = 1∕x, on [−3, 0) ∪ (0, 3].
Solution From the graph in Figure 4.27, f clearly fails to have either an absolute
2 maximum or an absolute minimum on [−3, 0) ∪ (0, 3]. The following table of values
for f(x) for x close to 0 suggests the same conclusion.
-3
x
3 x 1/x x 1/x
1 1 −1 −1
0.1 10 −0.1 −10
0.01 100 −0.01 −100
0.001 1000 −0.001 −1000
FIGURE 4.27 0.0001 10,000 −0.0001 −10,000
y = 1∕x 0.00001 100,000 −0.00001 −100,000
0.000001 1,000,000 −0.000001 −1,000,000

The most obvious difference between the functions in examples 3.1 and 3.2 is that
f(x) = 1∕x is not continuous throughout the interval [−3, 3]. We offer the following
theorem without proof.

THEOREM 3.1 (Extreme Value Theorem)
A continuous function f defined on a closed, bounded interval [a, b] attains both an
absolute maximum and an absolute minimum on that interval.

While you do not need to have a continuous function or a closed interval to have
an absolute extremum, Theorem 3.1 says that continuous functions are guaranteed to
have an absolute maximum and an absolute minimum on a closed, bounded interval.
In example 3.3, we revisit the function from example 3.2, but look on a different
interval.

y
EXAMPLE 3.3 Finding Absolute Extrema of a Continuous Function

1 Find the absolute extrema of f(x) = 1∕x on the interval [1, 3].
Solution Notice that on the interval [1, 3], f is continuous. Consequently, the Ex-
treme Value Theorem guarantees that f has both an absolute maximum and an ab-
x solute minimum on [1, 3]. Judging from the graph in Figure 4.28, it appears that f(x)
1 3 reaches its maximum value of 1 at x = 1 and its minimum value of 1∕3 at x = 3.
FIGURE 4.28 Our objective is to determine how to locate the absolute extrema of a given func-
y = 1∕x on [1, 3] tion. Before we do this, we need to consider an additional type of extremum.

DEFINITION 3.2
(i) f(c)isa local maximum of f if f(c) ≥ f(x) for all x in some open interval
Copyright © McGraw-Hill Education (ii) f(c)isa local minimum of f if f(c) ≤ f(x) for all x in some open interval
containing c.
containing c.

In either case, we call f(c)a local extremum of f.

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