Answer
See the explanation
Work Step by Step
Hypotheses of Rolle's Theorem:
1. The function \( f(x) \) is continuous on the closed interval \([a, b]\).
2. The function \( f(x) \) is differentiable on the open interval \((a, b)\).
3. \( f(a) = f(b) \).
Conclusion of Rolle's Theorem:
There exists at least one number \( c \) in the open interval \((a, b)\) such that \( f'(c) = 0 \).
The hypotheses of Rolle's Theorem are indeed necessary. Let's consider each one:
1. Continuity on the closed interval ensures that there are no "jumps" or "holes" in the function within the interval. This condition guarantees that the function can be drawn without lifting the pencil, which is crucial for the Intermediate Value Theorem to hold.
2. Differentiability on the open interval implies that the function has a well-defined derivative at every point within the interval. This condition is essential for ensuring that the tangent lines to the function exist at every point within the interval.
3. \( f(a) = f(b) \) implies that the function has the same value at both endpoints of the interval. This condition essentially establishes the requirement that the function starts and ends at the same height, which is crucial for the geometric interpretation of Rolle's Theorem.
Without these hypotheses, the conclusion of Rolle's Theorem may not hold. For instance, if the function is not continuous, it may have jumps or discontinuities that prevent the existence of a point where the derivative is zero. Similarly, if the function is not differentiable, it may have sharp corners or vertical tangents where the derivative does not exist. And if \( f(a) \neq f(b) \), there may not be a point where the function ends at the same height as it started, which is fundamental for the application of Rolle's Theorem. Therefore, the hypotheses are necessary to ensure the validity of the conclusion.
