Let `f:[0,2] to RR` be the function defined by `f(x)=(3-sin (2pix)) sin (pi x-(pi)/(4)) - sin (3pi x+(pi)/(4))`
If `alpha,beta in [0,2]` are such that `{x in [0,2]: f(x) ge0}= [alpha, beta]`, then the value of `beta- alpha` is._________

Answer

To solve the problem, we need to analyze the function \( f(x) = (3 - \sin(2\pi x)) \sin\left(\pi x - \frac{\pi}{4}\right) - \sin\left(3\pi x + \frac{\pi}{4}\right) \) and find the interval \([ \alpha, \beta ]\) where \( f(x) \geq 0 \).

### Step-by-Step Solution:

1. **Substitution**:
Let's set \( \theta = \pi x - \frac{\pi}{4} \). Then, we can express \( x \) in terms of \( \theta \):
\[ x = \frac{\theta + \frac{\pi}{4}}{\pi} \]
The limits for \( x \) from 0 to 2 correspond to \( \theta \) ranging from \( -\frac{\pi}{4} \) to \( \frac{7\pi}{4} \).

2. **Rewriting the Function**:
Substitute \( \theta \) into the function:
\[ f(x) = (3 - \sin(2\theta + \frac{\pi}{2})) \sin(\theta) - \sin(3(\frac{\theta + \frac{\pi}{4}}{\pi}) + \frac{\pi}{4}) \]
This simplifies to:
\[ f(x) = (3 - \cos(2\theta)) \sin(\theta) - \sin(3\theta + \frac{3\pi}{4}) \]

3. **Analyzing \( f(x) \geq 0 \)**:
We need to find where \( f(x) \geq 0 \). This means we need to analyze the expression:
\[ (3 - \cos(2\theta)) \sin(\theta) - \sin(3\theta + \frac{3\pi}{4}) \geq 0 \]

4. **Finding Critical Points**:
The critical points occur when \( \sin(\theta) = 0 \) or when \( 3 - \cos(2\theta) = 0 \).

- For \( \sin(\theta) = 0 \):
\[ \theta = n\pi \quad (n \in \mathbb{Z}) \]
The relevant values in the range \( [-\frac{\pi}{4}, \frac{7\pi}{4}] \) are \( 0, \pi \).

- For \( 3 - \cos(2\theta) = 0 \):
\[ \cos(2\theta) = 3 \quad \text{(not possible since } \cos \text{ ranges from -1 to 1)} \]

5. **Evaluating \( f(x) \) at Critical Points**:
Check the values of \( f(x) \) at the critical points \( \theta = 0 \) and \( \theta = \pi \):
- At \( \theta = 0 \):
\[ f(0) = (3 - 1)(0) - \sin(\frac{3\pi}{4}) = 0 - \frac{\sqrt{2}}{2} < 0 \]
- At \( \theta = \pi \):
\[ f(\pi) = (3 - (-1))(0) - \sin(\frac{3\pi}{4}) = 0 - \frac{\sqrt{2}}{2} < 0 \]

6. **Finding Intervals**:
We need to check the intervals between these critical points to find where \( f(x) \geq 0 \).

By testing values in the intervals:
- For \( \theta \) in \( (0, \pi) \), we find that \( f(x) \) is non-negative.

7. **Finding \( \alpha \) and \( \beta \)**:
From the analysis, we find:
\[ \alpha = \frac{1}{4}, \quad \beta = \frac{5}{4} \]

8. **Calculating \( \beta - \alpha \)**:
\[ \beta - \alpha = \frac{5}{4} - \frac{1}{4} = 1 \]

### Final Answer:
The value of \( \beta - \alpha \) is \( \boxed{1} \).