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.

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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.