Bernoulli's Rule: Existence of a Point c in (a,b)

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In summary, Rudin explains in his treatment of L'hôpital/Bernoulli's rule that if the differentiable quotient f'(x)/g'(x) approaches a limit A as x approaches a, and A is less than r, then there exists a point c between a and b such that for all x between a and c, f'(x)/g'(x) is less than r. This is because the theorem is not just about the existence of a single x in the interval, but rather about all x in that interval satisfying the given condition.
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Bachelier
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In his treatment of L'hôpital/Bernoulli's rule (please see attached), Rudin before ineq. ## (17)## mentions that since the differentiable quotient

##\frac{f'(x)}{g'(x)} \rightarrow A## as ##x \rightarrow a## and ##A<r## then there exists a pt ##c \in (a,b) \ s.t. \ a<x<c \Rightarrow \ \frac{f'(x)}{g'(x)}<r##

Is it so because ##x## approaches ##a## that's why he used ##a<x<c## instead of ##c<x<b##

and why this ##c## in the first place? What's wrong with just saying, ##\exists x \in (a,b)## etc
 

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The theorem is not there exists an x in the interval, but rather for all x, a < x <c, f'/g' < r.
 
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mathman said:
The theorem is not there exists an x in the interval, but rather for all x, a < x <c, f'/g' < r.

True. Thanks
 

Related to Bernoulli's Rule: Existence of a Point c in (a,b)

1. What is Bernoulli's Rule and what does it state?

Bernoulli's Rule is a mathematical theorem that states that for a continuous function f defined on the closed interval [a,b], if f(a) and f(b) have opposite signs, then there exists at least one point c in the interval such that f(c) = 0.

2. How is Bernoulli's Rule used in mathematics?

Bernoulli's Rule is often used to prove the existence of roots for continuous functions. It is also a key component in many other mathematical proofs and theorems.

3. Can Bernoulli's Rule be applied to all functions?

No, Bernoulli's Rule can only be applied to continuous functions on a closed interval [a,b]. It does not work for discontinuous or non-continuous functions.

4. What is the significance of the point c in Bernoulli's Rule?

The point c represents the location of the root of the function f(x). This point is important because it proves the existence of at least one root for the function, which can have many real-world applications.

5. How does Bernoulli's Rule relate to the Intermediate Value Theorem?

Bernoulli's Rule is a special case of the Intermediate Value Theorem, which states that for a continuous function on a closed interval, if a and b are two points in the interval and f(a) and f(b) have opposite signs, then there exists at least one point c between a and b where f(c) = 0.

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