Question

Let \( f: \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that \( f(0) = 0 \), \( f(2) = 4 \), \( f(4) = 4 \), and \( f(8) = 12 \). Then which of the following statements is/are correct?

• A. \( f'(x) \leq 1 \) for all \( x \in [0, 2] \)
• B. \( f'(x_1) > 1 \) for some \( x_1 \in [0, 2] \)
• C. \( f'(x_2) > 1 \) for some \( x_2 \in [4, 8] \)
• D. \( f''(x_3) = 0 \) for some \( x_3 \in [0, 8] \)

Answer 1

The Correct Option is B, C, D

The correct option is (B): \( f'(x_1) > 1 \) for some \( x_1 \in [0, 2] \) , (C): \( f'(x_2) > 1 \) for some \( x_2 \in [4, 8] \), (D): \( f''(x_3) = 0 \) for some \( x_3 \in [0, 8] \)