Let[a,b]R be such that f' * (x) = 0 cach point in (a,b). Prove that f is constant in [a,b]

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

### Proof:
1. **Given:**
   - \( f'(x) = 0 \) for all \( x \in (a, b) \).

2. **By the Mean Value Theorem (MVT):**
   - The MVT states that if a function \( f \) is continuous on a closed interval \( [c, d] \) and differentiable on the open interval \( (c, d) \), then there exists a point \( \xi \in (c, d) \) such that:
     \[
     f'(\xi) = \frac{f(d) - f(c)}{d - c}.
     \]
   - In this case, we know that \( f'(x) = 0 \) for all \( x \in (a, b) \), which means that \( f'(\xi) = 0 \) for any \( \xi \in (a, b) \).

3. **Apply the MVT to any subinterval of \( [a, b] \):**
   - Take any points \( c \) and \( d \) such that \( a \leq c < d \leq b \). The MVT guarantees a point \( \xi \in (c, d) \) where \( f'(\xi) = \frac{f(d) - f(c)}{d - c} \).
   - But since \( f'(x) = 0 \) for all \( x \in (a, b) \), we must have:
     \[
     \frac{f(d) - f(c)}{d - c} = 0,
     \]
     which implies that \( f(d) = f(c) \).

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

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

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.

Let [a,b] R be such that f'(x) = 0 at each point in (a,b). Prove that f is constant in [a,b].

Explore the proof that if f'(x) = 0 at every point in (a, b), then f is constant on [a, b]. This solution applies the Mean Value Theorem and basic calculus principles, suitable for advanced high school students in Grades 11-12.

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