Application of Derivatives
Mean Value Theorem
IIT-JEE 1992
Grade 12
Question:
The Lagrange's Mean Value Theorem is NOT applicable to $f(x) = x^{1/3}$ on $[-1, 1]$ because:
(1) $f$ is not continuous on $[-1, 1]$
(2) $f$ is not bounded on $[-1, 1]$
(3) $f$ is not differentiable at $x = 0$
(4) $f(-1) \neq f(1)$
Step-by-Step Solution
Key Concept: Examine the behaviour of $f'(x) = \frac{1}{3}x^{-2/3}$ as $x \to 0$ and check which of the LMVT hypotheses this violates.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (3)