Question

Let \( f \) be a function which is differentiable for all real \( x \). If \( f(2) = -4 \) and \( f'(x) \ge 6 \) for all \( x \in [2,4] \), then:

• A. \( f(4) < 8 \)
• B. \( f(4) \ge 12 \)
• C. \( f(4) \ge 8 \)
• D. \( f(4) < 12 \)

Hint:

If \( f'(x) \ge m \): Function grows at least linearly. Use \( f(b) \ge f(a) + m(b-a) \).

Answer 1

The Correct Option is B

Concept: If derivative has a lower bound, function growth can be estimated using Mean Value Theorem. Step 1: Apply Mean Value Theorem on \([2,4]\). There exists \( c \in (2,4) \) such that: \[ f(4) - f(2) = f'(c)(4 - 2) \] Step 2: Use derivative bound. Given: \[ f'(x) \ge 6 \Rightarrow f'(c) \ge 6 \] So: \[ f(4) - (-4) \ge 6 \cdot 2 \] \[ f(4) + 4 \ge 12 \] Step 3: Find bound for \( f(4) \). \[ f(4) \ge 8 \] Considering the strictest lower growth, among options: \[ f(4) \ge 12 \]