Lecture 15 max min - section 4.2

Maximum 4.2 and Minimum Values

2
Maximums & Minimums
Absolute also known as “Global”
Relative also know as “Local”

3
Example 1 – A Function with Infinitely Many Extreme Values
The function f (θ) = cos θ takes on its (local and absolute)
maximum value of 1 infinitely many times, since
cos 2np = 1 for any integer n and –1 £ cos θ £ 1 for all θ.
Likewise, cos(2n + 1)p = –1 is its minimum value, where n is
any integer.

5
Maximum and Minimum Values
Figures 8 and 9 show that a function need not possess
extreme values if either hypothesis (continuity or closed
interval) is omitted from the Extreme Value Theorem.
Figure 8 This function has minimum value Figure 9
f (2) = 0, but no maximum value.
This continuous function g has
no maximum or minimum.

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The function f whose graph is shown in Figure 8 is defined
on the closed interval [0, 2] but has no maximum value.
[Notice that the range of f is [0, 3). The function takes on
values arbitrarily close to 3, but never actually attains the
value 3.]
This does not contradict the Extreme Value Theorem
because f is not continuous.

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The function g shown in Figure 9 is continuous on the open
interval (0, 2) but has neither a maximum nor a minimum
value. [The range of g is (1, ). The function takes on
arbitrarily large values.]
This does not contradict the Extreme Value Theorem
because the interval (0, 2) is not closed.

8
Finding Max/Min Values
Figure 10

10
If f (x) = x3, then f ¢(x) = 3x2, so f ¢(0) = 0.
But f has no maximum or minimum at 0, as you can see
from its graph in Figure 11.
Figure 11
If f (x) = x3, then f ¢(0) = 0 but ƒ
has no maximum or minimum.

The fact that f ¢(0) = 0 simply means that the curve y = x3 has
a horizontal tangent at (0, 0).
11
Instead of having a maximum or minimum at (0, 0), the
curve crosses its horizontal tangent there.
Thus, when f ¢(c) = 0, f doesn’t necessarily have a maximum
or minimum at c. (In other words, the converse of Fermat’s
Theorem is false in general.)

There may be an extreme value where f ¢(c) does not exist.
For instance, the function f (x) = | x | has its (local and
absolute) minimum value at 0 (see Figure 12), but that value
cannot be found by setting f ¢(x) = 0 because, f ¢(0) does not
exist.
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Figure 12
If f (x) = | x |, then f (0) = 0 is a
minimum value, but f ¢(0) does not exist.

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Fermat’s Theorem does suggest that we should at least
start looking for extreme values of f at the numbers c where
f ¢(c) = 0 or where f ¢(c) does not exist. Such numbers are
given a special name.
In terms of critical numbers, Fermat’s Theorem can be
rephrased as follows.

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Example – Findind Critical #’s

15
Exercise – Critical #’s

Finding Max/Min Values
To find an absolute maximum or minimum of a continuous
function on a closed interval, we note that either it is local or
it occurs at an endpoint of the interval.
Thus the following three-step procedure always works.
16

17
Example – Finding Max/Min

18
Exercises – Finding Max/Min

Lecture 15 max min - section 4.2

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Lecture 15 max min - section 4.2