The problem asks to find the minimum positive period of the function $f(x) = 4\sin(4x + \frac{\pi}{5})$.

1. Problem Description

The problem asks to find the minimum positive period of the function $f(x) = 4\sin(4x + \frac{\pi}{5})$.

2. Solution Steps

The general form of a sine function is $f(x) = A\sin(Bx + C)$, where $A$ is the amplitude, $B$ affects the period, and $C$ is the phase shift. The period $T$ of the function is given by the formula:

$T = \frac{2\pi}{|B|}$

In our case, the function is $f(x) = 4\sin(4x + \frac{\pi}{5})$, so $B = 4$.

Therefore, the period is:

$T = \frac{2\pi}{|4|} = \frac{2\pi}{4} = \frac{\pi}{2}$

3. Final Answer

The minimum positive period of the function is $\frac{\pi}{2}$.