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: To solve $f'(x) < 2f(x)$, multiply both sides by $e^{-2x}$ to obtain $e^{-2x}f'(x) - 2e^{-2x}f(x) < 0$, which is the derivative of the product $e^{-2x}f(x)$. This transforms the inequality into $\frac{d}{dx}[e^{-2x}f(x)] < 0$, meaning $e^{-2x}f(x)$ is strictly decreasing on the domain.
Given $f'(x) 0$ preserves the inequality direction, and fail to recognize that this creates a derivative of a product.
Correct Answer: 8