Math Problem Statement
Let[a,b]R be such that f' * (x) = 0 cach point in (a,b). Prove that f is constant in [a,b]
Solution
We are given that for each point in the interval , and we need to prove that is constant on the closed interval .
Proof:
-
Given:
- for all .
-
By the Mean Value Theorem (MVT):
- The MVT states that if a function is continuous on a closed interval and differentiable on the open interval , then there exists a point such that:
- In this case, we know that for all , which means that for any .
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Apply the MVT to any subinterval of :
- Take any points and such that . The MVT guarantees a point where .
- But since for all , we must have: which implies that .
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Conclusion:
- Since for any and , this means that takes the same value at any two points in .
- Therefore, is constant on .
Thus, we have proven that if for each point in , then must be constant on .
Would you like further details or clarifications?
Related Questions:
- What are the assumptions of the Mean Value Theorem?
- Can a function be constant in an interval but have a non-zero derivative at a point?
- What are the differences between local and global extrema?
- How do we apply the Mean Value Theorem to prove other properties of functions?
- What happens if the derivative is zero at isolated points instead of throughout the interval?
Tip: The Mean Value Theorem is a powerful tool for analyzing differentiable functions, especially when studying behavior over intervals.
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Math Problem Analysis
Mathematical Concepts
Calculus
Differentiation
Continuity
Formulas
f'(x) = 0 for all x in (a, b)
Mean Value Theorem formula: f'(ξ) = (f(d) - f(c)) / (d - c)
Theorems
Mean Value Theorem
Suitable Grade Level
Grades 11-12
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