Let `f(x)={b^(2)+(a-1)b-(1)/(4)}x+int_(0)^(x)(sin^(4)theta+cos^(4)theta)d theta` If `f(x)` be a non- decreasing function `AA x in R` and `AA b in R` then "a" can belong to

Answer

To solve the problem, we need to analyze the function \( f(x) \) and determine the conditions under which it is a non-decreasing function.

### Step-by-Step Solution:

1. **Define the function**:
\[ f(x) = \left(b^2 + (a-1)b - \frac{1}{4}\right)x + \int_{0}^{x} \left(\sin^4 \theta + \cos^4 \theta\right) d\theta \]

2. **Differentiate \( f(x) \)**:
To find if \( f(x) \) is non-decreasing, we need to compute its derivative \( f'(x) \):
\[ f'(x) = b^2 + (a-1)b - \frac{1}{4} + \left(\sin^4 x + \cos^4 x\right) \]
Here, the derivative of the integral \( \int_{0}^{x} g(\theta) d\theta \) is simply \( g(x) \) due to the Fundamental Theorem of Calculus.

3. **Simplify \( \sin^4 x + \cos^4 x \)**:
We can simplify \( \sin^4 x + \cos^4 x \) using the identity:
\[ \sin^4 x + \cos^4 x = (\sin^2 x + \cos^2 x)^2 - 2\sin^2 x \cos^2 x = 1 - 2\sin^2 x \cos^2 x \]
Thus,
\[ f'(x) = b^2 + (a-1)b - \frac{1}{4} + 1 - 2\sin^2 x \cos^2 x \]

4. **Set the condition for non-decreasing**:
For \( f(x) \) to be non-decreasing, we require:
\[ f'(x) \geq 0 \]
This leads to:
\[ b^2 + (a-1)b + \frac{3}{4} - 2\sin^2 x \cos^2 x \geq 0 \]

5. **Analyze the term \( -2\sin^2 x \cos^2 x \)**:
The maximum value of \( \sin^2 x \cos^2 x \) is \( \frac{1}{4} \), so:
\[ -2\sin^2 x \cos^2 x \geq -\frac{1}{2} \]
Therefore, we can write:
\[ b^2 + (a-1)b + \frac{3}{4} - \frac{1}{2} \geq 0 \]
Simplifying gives:
\[ b^2 + (a-1)b + \frac{1}{4} \geq 0 \]

6. **Find the conditions on \( a \)**:
The quadratic \( b^2 + (a-1)b + \frac{1}{4} \) must be non-negative for all \( b \). This occurs when the discriminant is less than or equal to zero:
\[ (a-1)^2 - 4 \cdot 1 \cdot \frac{1}{4} \leq 0 \]
Simplifying gives:
\[ (a-1)^2 - 1 \leq 0 \]
This can be factored as:
\[ (a-2)(a) \leq 0 \]

7. **Determine the intervals for \( a \)**:
The solution to the inequality \( (a-2)(a) \leq 0 \) gives us:
\[ a \in [0, 2] \]

### Final Answer:
Thus, the values that \( a \) can belong to are:
\[ \boxed{[0, 2]} \]