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 f(x)=0f'(x) = 0 for each point in the interval (a,b)(a, b), and we need to prove that ff is constant on the closed interval [a,b][a, b].

Proof:

  1. Given:

    • f(x)=0f'(x) = 0 for all x(a,b)x \in (a, b).
  2. By the Mean Value Theorem (MVT):

    • The MVT states that if a function ff is continuous on a closed interval [c,d][c, d] and differentiable on the open interval (c,d)(c, d), then there exists a point ξ(c,d)\xi \in (c, d) such that: f(ξ)=f(d)f(c)dc.f'(\xi) = \frac{f(d) - f(c)}{d - c}.
    • In this case, we know that f(x)=0f'(x) = 0 for all x(a,b)x \in (a, b), which means that f(ξ)=0f'(\xi) = 0 for any ξ(a,b)\xi \in (a, b).
  3. Apply the MVT to any subinterval of [a,b][a, b]:

    • Take any points cc and dd such that ac<dba \leq c < d \leq b. The MVT guarantees a point ξ(c,d)\xi \in (c, d) where f(ξ)=f(d)f(c)dcf'(\xi) = \frac{f(d) - f(c)}{d - c}.
    • But since f(x)=0f'(x) = 0 for all x(a,b)x \in (a, b), we must have: f(d)f(c)dc=0,\frac{f(d) - f(c)}{d - c} = 0, which implies that f(d)=f(c)f(d) = f(c).
  4. Conclusion:

    • Since f(d)=f(c)f(d) = f(c) for any cc and d[a,b]d \in [a, b], this means that ff takes the same value at any two points in [a,b][a, b].
    • Therefore, ff is constant on [a,b][a, b].

Thus, we have proven that if f(x)=0f'(x) = 0 for each point in (a,b)(a, b), then ff must be constant on [a,b][a, b].

Would you like further details or clarifications?

Related Questions:

  1. What are the assumptions of the Mean Value Theorem?
  2. Can a function be constant in an interval but have a non-zero derivative at a point?
  3. What are the differences between local and global extrema?
  4. How do we apply the Mean Value Theorem to prove other properties of functions?
  5. 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