Question

If $f(x)$ is a twice differentiable function such that $f(0) = f(1) = f'(0) = 0$, then

• A. \(f''(x) \neq 0 \ \forall \ x \in (0,1)\)
• B. \(f''(x) = 0 \ \forall \ x \in (0,1)\)
• C. \(f(x)\) is a constant function
• D. \(f''(x) = 0\) for some \(x \in (0,1)\)

Hint:

Twice application of Rolle’s Theorem can help confirm the existence of points where the second derivative vanishes.

Answer 1

The Correct Option is D

Given: \[ f(0) = f(1) = 0, \quad f'(0) = 0 \] By Rolle’s Theorem applied on \([0,1]\), since \(f(0) = f(1)\), and \(f(x)\) is differentiable, there exists \(c \in (0,1)\) such that \(f'(c) = 0\). Now since \(f'(0) = 0\), apply Rolle’s Theorem again to \(f'\) on \([0,c]\) to get \(f''(x) = 0\) for some \(x \in (0,c) \subset (0,1)\).