What sets does the Mean Value Theorem apply to?

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SUMMARY

The Mean Value Theorem (MVT) applies specifically to real numbers a and b, which must be elements of the closed interval [a, b] where the function f(x) is continuous. The theorem asserts a universal claim that for every pair of points a and b within this interval, there exists at least one point c in the open interval (a, b) such that f'(c) equals the average rate of change of the function over [a, b]. This conclusion is derived from the hypotheses that f(x) is continuous on [a, b] and differentiable on (a, b).

PREREQUISITES
  • Understanding of the Mean Value Theorem
  • Knowledge of function continuity and differentiability
  • Familiarity with closed and open intervals in real analysis
  • Basic calculus concepts, particularly derivatives
NEXT STEPS
  • Study the formal proof of the Mean Value Theorem
  • Explore applications of the Mean Value Theorem in real-world scenarios
  • Learn about the implications of the Mean Value Theorem in optimization problems
  • Investigate related theorems, such as Rolle's Theorem
USEFUL FOR

Students of calculus, mathematics educators, and anyone seeking to deepen their understanding of the Mean Value Theorem and its applications in real analysis.

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Homework Statement



What set(s) are a and b assumed to be elements of? Does the mean Value Theorem make a universal claim (for all a ... for all b) or an existential claim (there exists a ... there exists b) about a and b? Explain how you came to this conclusion.


Homework Equations



We were told to look at the wikipedia page:

http://en.wikipedia.org/wiki/Mean_value_theorem

The Attempt at a Solution



Initially I thought that a and b are assumed to be elements of the domain of the function in question... so assumed that f(x) fulfills the antecedent of the theorem, then for every a, b in the domain of f(x) the consequent is true.

But then can I also assume that a and b are all elements of the real numbers, then for every function that happens to be continuous on this interval, the consequent is true?

Does that matter? Am I confusing myself? lol.
 
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The hypotheses are that a and b are real numbers and f is a function such that:

f(x) is continuous on [a,b]
f'(x) exists on (a,b)

Does that help you think through your question?
 
LCKurtz said:
The hypotheses are that a and b are real numbers and f is a function such that:

f(x) is continuous on [a,b]
f'(x) exists on (a,b)

Does that help you think through your question?

Yeah I think you just answered my question.
 

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