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We now need only compare the values of f at the endpoints and the approximate
critical numbers:
f(a) ≈ 7.3, f(b) ≈−1.7, f(c) ≈−1.3
f(d) ≈−4.3, f(−2) ≈−0.3 and f(2.5) ≈ 3.0.
Thus, the absolute maximum is approximately f(−1.26410884789) ≈ 7.3 and the
absolute minimum is approximately f(2.01830371473) ≈−4.3.
It is important (especially in light of how much of our work here was
approximate and graphical) to verify that the approximate extrema correspond
with what we expect from the graph of y = f(x). Since these correspond closely, we
have great confidence in their accuracy.
We have now seen how to locate the absolute extrema of a continuous function on
a closed interval. In section 4.4, we see how to find local extrema.

BEYOND FORMULAS

The Extreme Value Theorem is an important but subtle result. Think of it this way.
If the hypotheses of the theorem are met, you will never waste your time looking
for the maximum of a function that does not have a maximum. That is, the problem
is always solvable. The technique described in Remark 3.4 always works, as long as
there are only finitely many critical numbers.

EXERCISES .3
4

WRITING EXERCISES y
10
1. Using one or more graphs, explain why the Extreme Value
Theorem is true. Is the conclusion true if we drop the hy-
pothesis that f is a continuous function? Is the conclusion 5
true if we drop the hypothesis that the interval is closed?
x
2. Using one or more graphs, argue that Fermat’s Theorem is -4 -2 2 4
true. Discuss how Fermat’s Theorem is used. Restate the
theorem in your own words to make its use clearer. -5
3. Suppose that f(t) represents your elevation after t hours on
′
a mountain hike. If you stop to rest, explain why f (t) = 0. -10
Discuss the circumstances under which you would be at a
local maximum, local minimum or neither.
4. Mathematically, an if/then statement is usually strictly one- 2. f(x) = x 2 2 on (a) (−∞, 1) ∪ (1, ∞),
directional. When we say “If A, then B” it is generally not (x − 1)
the case that “If B, then A” is also true. When both are true, (b) (−1, 1), (c) (0, 1), (d) [−2, −1]
we say “A if and only if B,” which is abbreviated to “A iff
B.” Consider the statement, “If you stood in the rain, then
you got wet.” Is this true? How does this differ from its con- y
verse, “If you got wet, then you stood in the rain.” Apply this 10
logic to both the Extreme Value Theorem and Fermat’s The-
orem: state the converse and decide whether its conclusion 8
is sometimes true or always true.
6 4
Copyright © McGraw-Hill Education trema (if they exist) of the function on the given interval. ............................................................
In exercises 1 and 2, use the graph to locate the absolute ex-
2
1
1. f(x) =
on (a) (0, 1) ∪ (1, ∞),
x − 1
2
x
1 1
]
[
2
4
-2
-4
(b) (−1, 1), (c) (0, 1), (d) − ,
2 2
257