minimum and maximum
There are two views on extreme values.
- The global maximum value is the greatest value of the function.
- The global minimum value is the least value of the function.
A function has at most, one maximum value and one minimum value, but they can occur at multiple domain numbers.
Global extrema are also called absolute extrema.
- A global maximum may occur at multiple domain numbers.
- A function can also not have a maximum value.
- A global minimum may occur at multiple domain numbers.
- A function can also not have a minimum value.
In contrast to a single maximum or minimum value, a function may also have values which are the greatest value in their own little neighborhood, but not the greatest overall value. We see these visually encoded as tops of hills and bottom of valleys on the graph. A function may have many of these local or relative extrema.
- is a local maximum value of the function , if there exists an such that
for all domain numbers, , within a distance of from .
- is a local minimum value of the function , if there exists an such that for all domain numbers, , within a distance of from .
Local extrema are also called relative extrema.
By default, all global extrema are automatically local extrema.
Let be a function. The graph of is displayed below.
- has no global maximum.
- The global minimum of is , which occurs at .
- has a local minimum of , which occurs at and a local minimum of , which occurs at .
- has a local maximum of , which occurs at
Let’s run through the idea of a local extrema for the previous example.
has a local minimum of , which occurs at .
Let . Then, .
for all in the domain within a distance of from , which would be the interval
.
has a local maximum of , which occurs at .
Let . Then, .
for all in the domain within a distance of from , which would be the interval
.
has a local minimum of , which occurs at .
Let . Then, .
for all in the domain within a distance of from , which would be the interval .
Note: As the previous example illustrates, global extrema are also local extrema.
Note: We picked , , and for , but we could have selected any small, positive numbers
that work.
Let be a function. The graph of is displayed below.
- has no global maximum.
- The global minimum of is , which occurs at .
- has a local minimum of , which occurs at and a local minimum of , which occurs at .
- has a local maximum of , which occurs at
Let . for all in the domain within a distance of from , which would be the interval .
Let . for all in the domain within a distance of from , which would be the interval .
Let . for all in the domain within a distance of from , which would be the interval .
Let be a function. The graph of is displayed below.
- has no global maximum.
- has no global minimum.
- has a local minimum of , which occurs at .
- has a local minimum of , which occurs at .
- has a local maximum of , which occurs at .
It appears that would have been the global maximum of , however, the point is missing from the graph and is not in the range. And, there is no real number “just below ”. If you choose any real number, , below to be the possible candidate for global maximum, there is the number . This number is in the range and . You cannot identify a specific number as the maximum, so there isn’t one.
Let . for all in the domain within a distance of from , which would be the interval .
Let . for all in the domain within a distance of from , which would be the interval .
Let . for all in the domain within a distance of from , which would be the interval .
Inside the natural numbers there is the idea of “next”. Each natural number has a
next number.
However, the real numbers does not have a “next” property. There is no “next” number for any real number.
Let be a real number.
Supposed is the “next” real number. Then we have a problem, because the real
number is between and . Therefore, cannot be the “next” real number. There is no
“next” real number.
A consequance of this is that each open interval contains an infinity number of
numbers.
Note: This is not true for closed intervals. For example, the closed interval contains only one number. This is one reason we talk about open intervals.
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more examples can be found by following this link
More Examples of Graphical Analysis