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 $f'(c) = 0$ is the statement of

• A. Lagrange’s mean value theorem
• B. Rolle’s Theorem
• C. Fundamental theorem of calculus
• D. Fundamental theorem of algebra

Hint:

If you see the condition $f(a) = f(b)$ leading to $f'(c) = 0$, it is always Rolle's Theorem. If $f(a) \neq f(b)$ leading to a non-zero slope, it is Lagrange's Mean Value Theorem.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem asks to identify a foundational theorem of differential calculus.
Detailed Explanation:
Let us define the theorems listed in the options:
- Lagrange's Mean Value Theorem: States that if $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists a $c \in (a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$.
- Rolle's Theorem: A special case of Lagrange's theorem. It states that if $f(a) = f(b)$ in addition to being continuous and differentiable, then there exists a $c \in (a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a} = 0$. This matches the problem statement exactly.
- Fundamental Theorem of Calculus: Relates differentiation and integration.
- Fundamental Theorem of Algebra: States that every non-zero, single-variable, complex polynomial of degree $n$ has exactly $n$ complex roots.

Step 2: Final Answer:
The given statement is Rolle's Theorem, matching Option (B).