Application of Derivatives
Rolle's Theorem
IIT-JEE 2004
Grade 12
Question:
For $f(x) = (x - 1)(x - 2)(x - 3)$ on $[1, 3]$, the number of values of $c \in (1, 3)$ satisfying Rolle's theorem is:
(1) $0$
(2) $1$
(3) $2$
(4) $3$
Step-by-Step Solution
Key Concept: Verify $f(1) = f(3) = 0$ first (so Rolle's theorem applies), then differentiate $f$, set $f'(x) = 0$, and count how many roots of this quadratic lie strictly inside $(1, 3)$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (3)