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

  • #1
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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.
 

Answers and Replies

  • #2
33,635
5,296

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
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
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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).
 

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