If $f'(x) < 2f(x)$ where $f: \left[\frac{1}{2}, 1\right] \to R$ such that $f\left(\frac{1}{2}\right) = 2e$ then maximum value of $f(\ln 2)$ is ____.
Step-by-Step Solution
Key Concept: Transform the inequality f'(x) < 2f(x) by multiplying both sides by e^(-2x) to obtain d/dx[e^(-2x)f(x)] < 0, which means e^(-2x)f(x) is strictly decreasing on [1/2, 1]. Use this monotonicity with the boundary condition f(1/2) = 2e to bound f(ln 2).
Given $f'(x) 0$ preserves the inequality direction, and fail to recognize that this creates a derivative of a product.
Correct Answer: 8