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

[Simkate]

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 !

 

[╔(σ_σ)╝]

What have you tried ?

Have you considered using intermediate value theorem along with induction ?

 

[Dick]

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

 

[╔(σ_σ)╝]

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.

 

[Dick]

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.

 

[╔(σ_σ)╝]

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.