` 0 le a le 3, 0 le b le 3` and the equation, `x^(2)+4+3 cos ( ax+b) = 2x` has atleast one solution, then find the value of (a+b) .

Answer

To solve the problem step by step, we need to analyze the given equation and the conditions provided.

### Step 1: Rewrite the Equation
The given equation is:
\[ x^2 + 4 + 3 \cos(ax + b) = 2x \]

We can rearrange this equation:
\[ x^2 - 2x + 4 + 3 \cos(ax + b) = 0 \]

### Step 2: Identify the Quadratic Function
Let’s define a function based on the rearranged equation:
\[ f(x) = x^2 - 2x + 4 + 3 \cos(ax + b) \]

### Step 3: Analyze the Quadratic Part
The quadratic part of the function \( x^2 - 2x + 4 \) can be analyzed. The vertex of this quadratic function can be found using the formula \( x = -\frac{b}{2a} \):
- Here, \( a = 1 \) and \( b = -2 \).
- The vertex \( x \) value is:
\[ x = \frac{2}{2} = 1 \]

Now, we can find the value of the quadratic at this vertex:
\[ f(1) = 1^2 - 2(1) + 4 + 3 \cos(a(1) + b) \]
\[ f(1) = 1 - 2 + 4 + 3 \cos(a + b) \]
\[ f(1) = 3 + 3 \cos(a + b) \]

### Step 4: Condition for At Least One Solution
For the equation \( f(x) = 0 \) to have at least one solution, the value of \( f(1) \) must be non-negative:
\[ 3 + 3 \cos(a + b) \geq 0 \]

This simplifies to:
\[ \cos(a + b) \geq -1 \]

### Step 5: Determine the Range of \( \cos(a + b) \)
Since \( \cos(a + b) \) can take values between -1 and 1, the condition \( \cos(a + b) \geq -1 \) is always satisfied. However, we need to find specific values of \( a \) and \( b \) such that the equation has at least one solution.

### Step 6: Find Maximum and Minimum Values
The maximum value of \( \cos(a + b) \) is 1 when \( a + b = 0 \) or multiples of \( 2\pi \). The minimum value is -1 when \( a + b = \pi \) or odd multiples of \( \pi \).

Given the constraints \( 0 \leq a \leq 3 \) and \( 0 \leq b \leq 3 \), we need to find combinations of \( a \) and \( b \) such that \( a + b \) is within the range that satisfies the equation.

### Step 7: Find Possible Values of \( a + b \)
The maximum value of \( a + b \) is \( 3 + 3 = 6 \). The minimum is \( 0 + 0 = 0 \).

To ensure \( f(1) \geq 0 \), we can set:
- \( a + b = \pi \) (approximately 3.14), which is within our range.

### Step 8: Conclusion
Thus, the value of \( a + b \) that satisfies the condition is:
\[ a + b = \pi \]

### Final Answer
The value of \( a + b \) is \( \pi \).