To solve the problem, we need to analyze the function \( f(x) = \max\left\{\frac{x}{n}, |\sin(\pi x)|\right\} \) and determine the points of non-differentiability for \( x \) in the interval \( (0, 4) \). We will also find out which values of \( n \) would not allow for the maximum number of non-differentiable points.
### Step-by-Step Solution:
1. **Understand the Functions Involved**:
- The function \( |\sin(\pi x)| \) oscillates between 0 and 1 with a period of 2. It reaches its maximum at \( x = 0, 1, 2, 3, \) and \( 4 \).
- The function \( \frac{x}{n} \) is a straight line that passes through the origin with a slope of \( \frac{1}{n} \).
2. **Graph the Functions**:
- The graph of \( |\sin(\pi x)| \) will have peaks at \( x = 0, 1, 2, 3, 4 \) and will be symmetric about these points.
- The line \( \frac{x}{n} \) will intersect the graph of \( |\sin(\pi x)| \) at various points depending on the value of \( n \).
3. **Determine Points of Intersection**:
- The points of non-differentiability occur where \( \frac{x}{n} = |\sin(\pi x)| \). These intersections create sharp corners in the graph of \( f(x) \).
- For \( f(x) \) to have maximum points of non-differentiability, the line \( \frac{x}{n} \) should intersect \( |\sin(\pi x)| \) at as many points as possible within the interval \( (0, 4) \).
4. **Finding Conditions for Intersections**:
- The maximum value of \( |\sin(\pi x)| \) is 1. Thus, for the line \( \frac{x}{n} \) to intersect the curve at \( x = 4 \), we need:
\[
\frac{4}{n} \leq 1 \implies n \geq 4
\]
- This means that \( n \) must be at least 4 for the line to intersect the curve at \( x = 4 \).
5. **Count Non-Differentiable Points**:
- If \( n = 4 \), the line intersects the curve at points \( x = 1, 2, 3, \) and \( 4 \), giving us 4 points of non-differentiability.
- If \( n > 4 \), the line will intersect the curve at more points, leading to more non-differentiable points.
6. **Evaluate the Options**:
- The options given are \( n = 4, 2, 5, 6 \).
- Since \( n \) must be at least 4 for maximum points of non-differentiability, \( n = 2 \) is not a valid option as it does not satisfy \( n \geq 4 \).
### Conclusion:
The value of \( n \) that cannot allow for maximum points of non-differentiability is \( \boxed{2} \).
