Question

Let \( f : \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that \[ f(0) = 0, \, f'(0) = 2, \, f(1) = -3. \] Then, which of the following is/are TRUE?

• A. \( |f'(x)| \leq 2 \) for all \( x \in [0, 1] \).
• B. \( |f'(x_1)|>2 \) for some \( x_1 \in [0, 1] \).
• C. \( |f''(x)|<10 \) for all \( x \in [0, 1] \).
• D. \( |f''(x_2)| \geq 10 \) for some \( x_2 \in [0, 1] \).

Hint:

Use the Mean Value Theorem (MVT) to analyze the behavior of derivatives and find points where the function's derivative exceeds a given bound. Similarly, use the rate of change to estimate second derivatives.

Answer 1

The Correct Option is B, D

Step 1: Analyze the information given.
We know that: \[ f(0) = 0, \, f'(0) = 2, \, f(1) = -3. \] Using the Mean Value Theorem (MVT) for the interval \( [0, 1] \), there exists some \( c \in (0, 1) \) such that: \[ f'(c) = \frac{f(1) - f(0)}{1 - 0} = \frac{-3 - 0}{1} = -3. \] Thus, the derivative \( f'(x) \) takes the value \( -3 \) for some \( x \in (0, 1) \). Step 2: Analyze the validity of each option. - **Option (A):** We have \( f'(0) = 2 \) and from the MVT, we know that \( f'(c) = -3 \) for some \( c \in (0, 1) \). Thus, \( f'(x) \) is not bounded by 2 in magnitude for the entire interval, so **option (A) is false**. - **Option (B):** Since \( f'(c) = -3 \) for some \( c \in (0, 1) \), this shows that \( |f'(x_1)|>2 \) for some \( x_1 \in [0, 1] \). Therefore, **option (B) is true**. - **Option (C):** To analyze the second derivative \( f''(x) \), consider that \( f'(x) \) varies significantly over the interval. By the MVT, the rate of change of \( f'(x) \) might lead to large values for \( f''(x) \). Given the steep change from \( f'(0) = 2 \) to \( f'(c) = -3 \), **option (C) is false**. - **Option (D):** Since the function \( f(x) \) experiences a large change in \( f'(x) \) within a short interval, the second derivative \( f''(x) \) must be large at some point in the interval. Hence, **option (D) is true**. Final Answer: \[ \boxed{(B) |f'(x_1)|>2 \text{ for some } x_1 \in [0, 1], \, (D) |f''(x_2)| \geq 10 \text{ for some } x_2 \in [0, 1].} \]