The question involves analyzing a mathematical statement based on the Mean Value Theorem (MVT) or Rolle's Theorem.

We are given:

• $f(x)$ is differentiable on $[a, b]$.
• $f(a) = b$ and $f(b) = a$.

The claim is that there exists a $c \in (a, b)$ such that $f'(c) = -1$.

Verification:

1. By the Mean Value Theorem (MVT), if $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$, there exists $c \in (a, b)$ such that: $f'(c) = \frac{f(b) - f(a)}{b - a}.$

2. Substituting $f(a) = b$ and $f(b) = a$ into the formula: $f'(c) = \frac{a - b}{b - a}.$

3. Simplify: $f'(c) = -1.$

This shows that the statement is TRUE because it directly follows from the Mean Value Theorem.

Would you like further clarification on the reasoning or additional examples?

Here are 5 related questions to explore further:

1. What are the conditions for the Mean Value Theorem to be applicable?
2. How does Rolle's Theorem relate to the Mean Value Theorem?
3. Can a function violate this statement if it's not differentiable on $(a, b)$?
4. What happens if $f(a) = f(b)$ in this context?
5. How can this theorem be applied to real-world problems involving rates of change?

Tip: Always verify the conditions of a theorem before applying it to ensure the function satisfies continuity and differentiability.