Is the supremum of [-r,r] always equal to r or -r?

In summary, the text says the function f(-r) is not f(r). However, if you take the supremum of f(x) not x, then this is indeed the function f(r).
  • #1
AKBAR
5
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Let 0 < r < 1. Then [tex]\sup_{x\in[-r,r]}f(x)=f(r)}[/tex], right? However, the text I'm reading says it's [tex]f(-r)[/tex]. How could this be? For example, say r = 0.5, then the least upper bound of [-0.5, 0.5] is 0.5, or r, right? I don't see how it could be -r. Thanks for any help.
 
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  • #2
Hi AKBAR! :smile:
AKBAR said:
Let 0 < r < 1. Then [tex]\sup_{x\in[-r,r]}f(x)=f(r)}[/tex], right?

No.

That will be true if f is increasing in [-r,r], but not in most other cases.

What is f? What is the relevance of 0 < r < 1? What is the context? :confused:
 
  • #3
AKBAR said:
Let 0 < r < 1. Then [tex]\sup_{x\in[-r,r]}f(x)=f(r)}[/tex], right? However, the text I'm reading says it's [tex]f(-r)[/tex]. How could this be? For example, say r = 0.5, then the least upper bound of [-0.5, 0.5] is 0.5, or r, right? I don't see how it could be -r. Thanks for any help.
You are taking the supremum ("least upper bound") of the values of f not x!

For example, if f(x)= x, then f(-r)= -r, f(r)= r and f takes on all values between -r and r. In that case the sup of f(x) on the interval (not the sup of the interval itself) is f(r).

But if f(x)= -x, then f(-r)= r, f(r)= -r and now the supremum occurs at -r: f(-r) is the largest value (in fact it is the maximum value).

And it can get more complicated than that. If f(x)= -x2, then f(-r)= f(r)= -r2< 0. The maximum (and so sup) occurs in the middle of the interval. The sup is f(0)= 0.
 
  • #4
Ahhhh...that really clears things up. Thank you guys.

The function in question was [tex]f(x)=\frac{(n-1)!}{(1+x)^n}[/tex]. So [tex]\sup_{-r\leq x\leq r}\frac{(n-1)!}{(1+x)^n}=\frac{(n-1)!}{(1-r)^n}[/tex] The r comes from the radius of convergence of a Taylor expansion (I'm reading about where T(x) = f(x) ).

Thanks again for the help.
 

What is the definition of supremum?

Supremum, also known as the least upper bound, is the smallest number that is greater than or equal to all the elements in a given set.

How is supremum different from maximum?

Supremum is the smallest upper bound of a set, while maximum is the largest number in a set. Maximum may or may not be equal to supremum, depending on the set.

What is the relationship between supremum and infimum?

Supremum and infimum are both types of bounds for a given set. Supremum is the smallest upper bound, while infimum is the largest lower bound.

What are some common examples of supremum in mathematics?

In a set of real numbers, the supremum would be the largest number that is still less than or equal to all other numbers in the set. In a set of functions, the supremum would be the function with the smallest upper bound.

Why is understanding supremum important in mathematics?

Supremum is a fundamental concept in mathematics, particularly in the study of real numbers and functions. It allows for the precise definition of important mathematical concepts such as continuity, convergence, and compactness.

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