Cauchy Mean Value Theorem Proof for Continuous and Integrable Functions

Iceman: The Cauchy Mean Value Theorem is a special case of the "Extended" Mean Value Theorem: if f is continuous on [a,b] and g is integrable and never 0 on [a,b] then there exists c in [a,b] such that: int(a to b) f(x)g(x)dx= f(c) int(a to b)g(x)dx. The proof is really rather simple: since g is never 0, g(x)>= epsilon>0. Then int(a to b) f(x)g(x)dx>= epsilon int(a to b) f(x)dx. Since f is continuous on [a,b], it
  • #1
iceman
Hi, I really need some help in sovling this proof!

Prove the Cauchy Mean Value Theorem:
If f,g : [a,b]->R satisfy f continuous, g integrable and
g(x)>=0 for all x then there exists element c is a member of set [a,b] so that
int(x=b,a)f(x)g(x)dx=f(c)int(x=b,a)g(x)dx.

Thanks for your help :D
 
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  • #2
Hi iceman,
imagine I'm a complete ignorant in mathematics. Then I can still type your key words into google, and get for instance this:

http://www.maths.abdn.ac.uk/~igc/tch/ma2001/notes/node42.html



Edit:
Sorry, couldn't help it. But if I was you, I'd ask my prof what's the good in posing problems the answer to which is in the literature. Can't he come up with something more creative?
 
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  • #3
Arcnets: You also have to be able to UNDERSTAND the results of a google search.
The "Cauchy Mean Value Theorem" your reference gives is clearly NOT the same as the "Cauchy Mean Value Theorem" in the original post. For one thing, the Cauchy Mean Value Theorem the OP asked about is an integral mean value theorem.

(I would use the phrase "extended mean value theorem" for the result given in Arcnet's link.)

Iceman: You can, however, USE the (extended) mean value theorem.

Let F(x)= int(t=a to x) f(t)g(t)dt and let G(x)= int(t= a to x)g(t)dt. Use those in the extended mean value theorem:

(F(b)- F(a))/(G(b)- G(a))= F'(c)/G'(c) for some c in [a,b].

That's not the complete answer- you will still need to do some work.
 
  • #4
OK,OK. Back to being helpful not provocative.
 
  • #5
Oh, darn! Provocative is so MUCH more fun!

(And yes, I have complained about doing peoples "google" work for them myself.)
 

What is the Cauchy Mean Value Theorem?

The Cauchy Mean Value Theorem is a fundamental theorem in calculus that states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point in the open interval where the slope of the tangent line is equal to the slope of the secant line connecting the endpoints of the interval.

What is the significance of the Cauchy Mean Value Theorem?

The Cauchy Mean Value Theorem is significant because it provides a way to relate the average rate of change of a function to the instantaneous rate of change at a specific point. This helps us understand the behavior of a function and make predictions about its values.

Can the Cauchy Mean Value Theorem be applied to all functions?

No, the Cauchy Mean Value Theorem can only be applied to functions that are continuous on a closed interval and differentiable on the open interval. If a function fails to meet these criteria, the theorem cannot be applied.

How is the Cauchy Mean Value Theorem different from the Mean Value Theorem?

The Mean Value Theorem is a special case of the Cauchy Mean Value Theorem, where the function is continuous on a closed interval and differentiable on the open interval. The Cauchy Mean Value Theorem is a more general form of the Mean Value Theorem, as it allows for the function to be discontinuous at the endpoints of the interval.

What are some applications of the Cauchy Mean Value Theorem?

The Cauchy Mean Value Theorem has various applications in calculus, including finding the maximum and minimum values of a function on an interval, proving the existence of solutions to differential equations, and determining the convergence of sequences and series.

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