It looks like you're working on the Rolle's Theorem, but the reasoning in your proof needs some clarification. Let's go over the proper steps to understand why your approach isn't quite right.

Statement of Rolle's Theorem

Rolle's Theorem states that if a function $f$ is continuous on a closed interval $[a, b]$ and differentiable on the open interval $(a, b)$, and if $f(a) = f(b)$, then there exists at least one point $c \in (a, b)$ such that: $f'(c) = 0.$

Your Proof

You started by assuming that a continuous function is bounded on the interval $I = [a, b]$, which is true. Every continuous function on a closed interval is indeed bounded (by the Extreme Value Theorem).

Then, you said there exist points $a, b \in I$ such that $f(a) \leq f(x) \leq f(b)$, but you didn't define which of these points are the maximum or minimum. This can cause confusion because we need to make sure that $f(a) = f(b)$, which is a necessary condition for applying Rolle's Theorem.

You seem to want to show that at least one point $c$ exists in $(a, b)$ where $f'(c) = 0$, but the argument you used doesn't follow from the requirements of the theorem.

Correct Proof Outline

1. Assumptions:

   • Let $f$ be continuous on $[a, b]$.
   • Let $f$ be differentiable on $(a, b)$.
   • Suppose $f(a) = f(b)$.

2. Existence of Extremum:

   • By the Extreme Value Theorem, since $f$ is continuous on the closed interval $[a, b]$, $f$ attains both a maximum and a minimum on this interval. Let's say that the maximum occurs at $x = c \in [a, b]$ and the minimum occurs at $x = d \in [a, b]$.

   • Now, if the maximum or minimum occurs at the endpoints $a$ or $b$, then either $f'(a) = 0$ or $f'(b) = 0$, or you can apply the Mean Value Theorem on the subintervals.

3. Use of the Mean Value Theorem:

   • The Mean Value Theorem tells us that if $f(a) = f(b)$, then there exists some point $c \in (a, b)$ such that $f'(c) = 0$.

What's Missing in Your Argument:

• You didn't show the condition that $f(a) = f(b)$, which is crucial for Rolle's Theorem to apply.
• The argument about finding the extremum (maximum or minimum) needs to be more formally tied to the endpoints or to the critical points within the interval.

Tip:

• Always remember that Rolle’s Theorem specifically requires $f(a) = f(b)$. Without this condition, you cannot guarantee the existence of a point where the derivative is zero.
• You may also want to recall the Mean Value Theorem, which plays a key role in deriving Rolle's Theorem.

Would you like more help with any part of the proof or clarification? Feel free to ask!

Here are 5 questions you can explore next:

1. What is the difference between Rolle's Theorem and the Mean Value Theorem?
2. Can Rolle's Theorem be applied to non-differentiable functions?
3. How can you visualize Rolle's Theorem graphically?
4. What happens if $f(a) \neq f(b)$?
5. How can you prove that the maximum or minimum occurs at the boundary of the interval?

Tip: The Extreme Value Theorem guarantees that continuous functions on closed intervals attain their maximum and minimum values at some points within the interval.