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

[SafiBTA]

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.

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.

 

[Mark44]

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.

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.

 

[Ray Vickson]

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.

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).