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.