Prove a function to be increasing using lmvt

To prove that a function is increasing using **Lagrange's Mean Value Theorem (LMVT)**, we need to go through the following steps:

### Lagrange's Mean Value Theorem (LMVT) Statement:
LMVT states that if a function \( f(x) \) is:
1. **Continuous** on a closed interval \([a, b]\),
2. **Differentiable** on the open interval \((a, b)\),

then there exists at least one point \( c \in (a, b) \) such that:

\[
f'(c) = \frac{f(b) - f(a)}{b - a}
\]

Now, if we can show that \( f'(x) \) is positive for all \( x \in (a, b) \), it implies that \( f(b) > f(a) \) for all \( a < b \), meaning the function is increasing.

### Steps to prove a function is increasing using LMVT:

1. **Check Continuity and Differentiability:**
   - Ensure the function \( f(x) \) is continuous on the closed interval \([a, b]\).
   - Ensure the function \( f(x) \) is differentiable on the open interval \((a, b)\).

2. **Apply LMVT:**
   By LMVT, for some \( c \in (a, b) \), we have:
   \[
   f'(c) = \frac{f(b) - f(a)}{b - a}
   \]
   The sign of \( f'(c) \) will indicate the behavior of \( f(x) \).

3. **Check the Derivative:**
   - If \( f'(x) > 0 \) for all \( x \in (a, b) \), then the slope of the secant line \( \frac{f(b) - f(a)}{b - a} \) will be positive for all \( a < b \), proving that \( f(x) \) is strictly increasing on \( [a, b] \).

### Example:
Let's consider the function \( f(x) = x^2 \), and prove it is increasing on the interval \( [1, 3] \).

1. **Continuity and Differentiability:**
   \( f(x) = x^2 \) is continuous and differentiable on all real numbers, so it's continuous on \( [1, 3] \) and differentiable on \( (1, 3) \).

2. **Apply LMVT:**
   By LMVT, there exists some \( c \in (1, 3) \) such that:
   \[
   f'(c) = \frac{f(3) - f(1)}{3 - 1} = \frac{9 - 1}{2} = 4
   \]
   So, for some \( c \in (1, 3) \), \( f'(c) = 4 \).

3. **Check the Derivative:**
   The derivative of \( f(x) = x^2 \) is \( f'(x) = 2x \), which is positive for all \( x > 0 \). Therefore, on the interval \( [1, 3] \), \( f'(x) > 0 \), which means \( f(x) \) is increasing on \( [1, 3] \).

Thus, we have proved using LMVT that \( f(x) = x^2 \) is increasing on the interval \( [1, 3] \).

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Would you like more details on this process or another example? 

Here are some related questions:
1. How does LMVT relate to the concept of monotonicity in functions?
2. Can LMVT be used to prove a function is decreasing?
3. What are the conditions for a function to be strictly increasing?
4. How does Rolle's Theorem differ from LMVT?
5. How do you prove a function is increasing without using LMVT?

**Tip:** To check whether a function is increasing or decreasing, always analyze the sign of its derivative.

Learn how to prove that a function is increasing using Lagrange's Mean Value Theorem (LMVT). This solution explains the key steps, including continuity, differentiability, and the application of the LMVT formula. Suitable for students in Grades 11-12.

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