Can a function have a local max but no global max?

In summary, it is possible for a function to have a local maximum value but no global maximum value. This can occur when a function is continuously increasing or decreasing without ever reaching a maximum or minimum value. A local maximum is a point on a function where the value of the function is greater than or equal to all nearby points on the function, but it may not be the highest point on the entire function. A global maximum is the highest value that a function reaches over its entire domain and can only occur at one point on the function. While it is not common, it is still important to consider when analyzing functions.
  • #1
SafiBTA
10
0

Homework Statement


I need to confirm if I correct in saying the following:
If f(x) is a function having the domain [a,b) as shown in the figure, then f(x) has several local maxima but none of them is global maximum, and f(x) does not have a global maximum.
H3dG3Jm.png


Homework Equations

and definitions[/B]
Global Maximum: f(c) is said to be the global maximum of a function f(x) if f(c)≥f(x) for all x on the domain of f(x).

The Attempt at a Solution


There exists no point c in the domain of f(x) such that f(c)≥f(x) for all x in the domain. Hence, although f(x) has several local maxima, f(x) does not have a global maximum.
 
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  • #2
SafiBTA said:

Homework Statement


I need to confirm if I correct in saying the following:
If f(x) is a function having the domain [a,b) as shown in the figure, then f(x) has several local maxima but none of them is global maximum, and f(x) does not have a global maximum.
H3dG3Jm.png


Homework Equations

and definitions[/B]
Global Maximum: f(c) is said to be the global maximum of a function f(x) if f(c)≥f(x) for all x on the domain of f(x).

The Attempt at a Solution


There exists no point c in the domain of f(x) such that f(c)≥f(x) for all x in the domain. Hence, although f(x) has several local maxima, f(x) does not have a global maximum.
Yes, the function in this graph has no global maximum.
 
  • #3
SafiBTA said:

Homework Statement


I need to confirm if I correct in saying the following:
If f(x) is a function having the domain [a,b) as shown in the figure, then f(x) has several local maxima but none of them is global maximum, and f(x) does not have a global maximum.
H3dG3Jm.png


Homework Equations

and definitions[/B]
Global Maximum: f(c) is said to be the global maximum of a function f(x) if f(c)≥f(x) for all x on the domain of f(x).

The Attempt at a Solution


There exists no point c in the domain of f(x) such that f(c)≥f(x) for all x in the domain. Hence, although f(x) has several local maxima, f(x) does not have a global maximum.
You are correct: the function has a (global) supremum, but not a global maximum on [a,b).
 

Related to Can a function have a local max but no global max?

1. Can a function have a local max but no global max?

Yes, it is possible for a function to have a local maximum value but no global maximum value. This means that there is a point on the function where it reaches a maximum value, but there is no maximum value that applies to the entire function.

2. How is this possible?

This can occur when a function is continuously increasing or decreasing without ever reaching a maximum or minimum value. In other words, the function does not have a finite endpoint or does not approach an asymptote.

3. What is a local maximum?

A local maximum is a point on a function where the value of the function is greater than or equal to all nearby points on the function. This point can be considered a peak in the graph of the function, but it may not be the highest point on the entire function.

4. What is a global maximum?

A global maximum is the highest value that a function reaches over its entire domain. It is the absolute maximum value of the function and can only occur at one point on the function.

5. Is it common for a function to have a local max but no global max?

No, it is not common for a function to have a local maximum value but no global maximum value. This is because most functions have a finite endpoint or approach an asymptote, allowing for a global maximum to exist. However, it is still possible and important to consider when analyzing functions.

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