Continuity and Differentiability
Rolle's Theorem
Premium Question
Grade 12
Question:
A function $f$ satisfies Rolle's theorem on $[a, b]$ with $c = \frac{a+b}{2}$. Which of the following is necessarily true?
(1) $f'(a) = f'(b)$
(2) $f'(c) = 0$
(3) $f(a) = f(b) = 0$
(4) $f$ is constant on $[a, b]$
Step-by-Step Solution
Key Concept: Rolle's theorem guarantees $f'(c) = 0$ for some $c \in (a, b)$. Here the specific value $c = \frac{a+b}{2}$ is given as the midpoint. The only statement that is necessarily true (given $c$ is the Rolle's point) is $f'(c) = 0$.
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Correct Answer: (2)