Proof of Existence of ξ in [a,b] for f(x_1) + f(x_2) +...+ f(x_n) / n

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Homework Help Overview

The discussion revolves around proving the existence of a point ξ in the interval [a,b] for a continuous function f, based on the average of function values at n points within that interval. The problem involves concepts from real analysis, particularly the properties of continuous functions and theorems related to them.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants suggest various approaches, including the use of the extreme value theorem and the intermediate value theorem. There is a discussion on whether induction is necessary for the proof, with differing opinions on its relevance.

Discussion Status

The discussion is ongoing, with participants exploring different theorems and their applicability to the problem. Some guidance has been offered regarding theorems that could be relevant, but there is no explicit consensus on the best approach or the necessity of induction.

Contextual Notes

Participants note the confusion of the original poster regarding how to start the problem and the implications of using different theorems. There is an acknowledgment of the need for clarity on the assumptions involved in the proof.

Simkate
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Let f be a continuous on the closed and bounded interval [a,b] and x_1, x_2, …, x_n ∈ [a,b]. Show that there necessarily exists ξ ∈ [a,b] such that:

f (ξ= [f(x_1) + f(x_2) + …f(x_n)] / n


How can I start this problem i am really confused! please help !
 
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What have you tried ?

Have you considered using intermediate value theorem along with induction ?
 


I would use the extreme value theorem first. Then use the intermediate value theorem. You can probably skip the induction.
 


Dick said:
I would use the extreme value theorem first. Then use the intermediate value theorem. You can probably skip the induction.
Extreme value theorem, why ? It doesn't matter I guess.

OP needs induction since we are not talking about some concrete sequence converging to something.
 


╔(σ_σ)╝ said:
Extreme value theorem, why ? It doesn't matter I guess.

OP needs induction since we are not talking about some concrete sequence converging to something.

If m<=f(xi)<=M, then you probably don't need induction to show sum f(xi)/n is between M and m.
 
Dick said:
If m<=f(xi)<=M, then you probably don't need induction to show sum f(xi)/n is between M and m.

Okay it's just a different approach. :-)

Induction seems more natural to me though.

Anyway, let me allow OP to do some thinking for him/herself.
 

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