Question

Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a function such that \( f(0) = 3 \) and \( |f'(x)| \leq 2 \), \( \forall x \in \mathbb{R} \). Then \( f(2) \) lies in which of the following intervals?

• A. \([-2, 5]\)
• B. \([0, 8]\)
• C. \([3, 8]\)
• D. \([-1, 7]\)

Hint:

Apply the Mean Value Theorem and bounds on the derivative to estimate function values.

Answer 1

The Correct Option is D

Since \( |f'(x)| \leq 2 \), the function is Lipschitz continuous with constant 2. By the Mean Value Theorem: \[ |f(2) - f(0)| \leq 2 \cdot |2 - 0| = 4 \] Given \( f(0) = 3 \), we have: \[ |f(2) - 3| \leq 4 \Rightarrow -4 \leq f(2) - 3 \leq 4 \Rightarrow -1 \leq f(2) \leq 7 \] So, \( f(2) \in [-1, 7] \).