. a) Let f: [a,b] Rhe such that f' * (x) >= 0 at each point in (a, b). Prove that f is monotonically. increasing in [a,b].

To prove that the function \( f: [a,b] \rightarrow \mathbb{R} \) is monotonically increasing on \( [a, b] \), given that \( f'(x) \geq 0 \) for every \( x \in (a, b) \), we can follow these steps:

### Step-by-Step Proof:

#### 1. **Definition of Monotonically Increasing**:
A function \( f \) is said to be monotonically increasing on an interval \( [a, b] \) if for every \( x_1, x_2 \in [a, b] \) such that \( x_1 < x_2 \), we have \( f(x_1) \leq f(x_2) \).

Thus, we need to show that \( f(x_1) \leq f(x_2) \) whenever \( x_1 < x_2 \), using the fact that \( f'(x) \geq 0 \) for all \( x \in (a, b) \).

#### 2. **Use of the Mean Value Theorem**:
The Mean Value Theorem (MVT) states that if a function \( f \) is continuous on a closed interval \( [x_1, x_2] \) and differentiable on the open interval \( (x_1, x_2) \), then there exists some \( c \in (x_1, x_2) \) such that:
\[
f'(c) = \frac{f(x_2) - f(x_1)}{x_2 - x_1}.
\]

Since \( f'(x) \geq 0 \) for all \( x \in (a, b) \), we have \( f'(c) \geq 0 \). Thus, applying the Mean Value Theorem for \( f \) on \( [x_1, x_2] \), we get:
\[
\frac{f(x_2) - f(x_1)}{x_2 - x_1} \geq 0.
\]

This implies that:
\[
f(x_2) - f(x_1) \geq 0,
\]
or equivalently:
\[
f(x_1) \leq f(x_2).
\]

#### 3. **Conclusion**:
Thus, we have shown that \( f(x_1) \leq f(x_2) \) for all \( x_1 < x_2 \) in \( [a, b] \). Therefore, \( f \) is monotonically increasing on \( [a, b] \).

### Conclusion:
The function \( f \) is monotonically increasing on \( [a, b] \), as required, because \( f'(x) \geq 0 \) for all \( x \in (a, b) \), and the Mean Value Theorem ensures that the function satisfies the condition of monotonicity.

---

Would you like further details or have any questions?

Here are 5 related questions to explore the concept further:
1. How does the Mean Value Theorem relate to differentiability?
2. Can you have a function where \( f'(x) = 0 \) everywhere but it is not constant?
3. If \( f'(x) \leq 0 \), what can be said about the function's monotonicity?
4. What happens if \( f'(x) \) is not defined at some points in \( (a, b) \)?
5. Can a function be increasing but not differentiable at certain points?

**Tip:** When applying the Mean Value Theorem, remember that the function must be continuous on the closed interval and differentiable on the open interval for the theorem to hold.

Let f: [a,b] → ℝ such that f'(x) ≥ 0 at each point in (a, b). Prove that f is monotonically increasing in [a,b].

This page explores how to prove that a function f is monotonically increasing on [a, b] given that f'(x) ≥ 0 for all x in (a, b). Using the Mean Value Theorem, this detailed explanation is suitable for college calculus students. Learn key concepts like differentiation and monotonicity in this proof-based approach.

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