Question

Let \( f: \mathbb{R} \to \mathbb{R} \) be a twice differentiable function. Then, which of the following statements is/are necessarily TRUE?

• A. \( f'' \) is continuous
• B. If \( f'(0) = f'(1) \), then \( f''(x) = 0 \) has a solution in \( (0,1) \)
• C. \( f' \) is bounded on \([8,10]\)
• D. \( f'' \) is bounded on \( (0,1) \)

Hint:

Whenever the derivative at two points is equal, Rolle’s theorem ensures the second derivative is zero somewhere between them.

Answer 1

The Correct Option is B, C

Step 1: Recall Rolle’s Theorem.
If a function \(f'(x)\) is continuous on \([a,b]\) and differentiable on \((a,b)\), and if \(f'(a) = f'(b)\), then there exists a point \(c \in (a,b)\) such that \(f''(c) = 0\).
Step 2: Apply the theorem to the given condition.
Given \(f\) is twice differentiable, \(f'\) is differentiable and hence continuous on \([0,1]\). Also, \(f'(0) = f'(1)\). Therefore, by Rolle’s theorem, there exists \(c \in (0,1)\) such that \(f''(c) = 0\).
Step 3: Examine other options.
(A) Continuity of \(f''\) is not guaranteed by twice differentiability; it only ensures \(f''\) exists.
(C) \(f'\) need not be bounded on an arbitrary interval without extra conditions.
(D) Similarly, \(f''\) may not be bounded on \((0,1)\).
Final Answer: \[ \boxed{(B)} \]