Question

Suppose $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$ . If $f(a) = f(b)$ , then there exists at least one number $c$ between $a$ and $b$ such that

• A. $f'(c) = 0$
• B. $f'(c) = b-a$
• C. $f'(c) = \lim_{x \to a} f(x)$
• D. $f'(c) = \lim_{x \to b} f(x)$

Hint:

Rolle's Theorem guarantees a horizontal tangent line ($f'(c) = 0$) when the endpoints of a smooth curve are at the same height ($f(a) = f(b)$).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question presents Rolle's Theorem, which is a fundamental theorem of calculus and a special case of the Mean Value Theorem.

Step 2: Detailed Explanation:
Rolle's Theorem states that if a real-valued function $f$ satisfies three conditions:
1. $f$ is continuous on the closed interval $[a, b]$.
2. $f$ is differentiable on the open interval $(a, b)$.
3. The functional values at the endpoints are equal, i.e., $f(a) = f(b)$.
Then, there must exist at least one number $c$ in the open interval $(a, b)$ such that the derivative of the function at that point is zero: \[ f'(c) = 0 \]
Geometrically, this means that for any continuous, differentiable curve that starts and ends at the same height, there is at least one point on the curve where the tangent line is horizontal (parallel to the X-axis).

Step 3: Final Answer:
There exists at least one number $c$ such that $f'(c) = 0$.