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

Answer: (B), (C), (D)

$f'(x_1) > 1$ for some $x_1 \in [0, 2]$

Answer 2

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 $x_3 \in [0, 8]$