Let `y = 2cos(sin^(2)theta + sintheta + 2)`. If range of y is [a, b] and S = {a, b}, then for some `x in S` (where [.] denotes the greatest integer function)

Options

A. if `x in I`, then `x lt 0`
B. if `xcancel(in)I,[cos^(-1)x]inS`
C. `t^(t)` is defined for some `tin[a,b]`
D. `x^(t//x)` is real for some `x in S`

Answer

To solve the problem, we need to find the range of the function \( y = 2 \cos(\sin^2 \theta + \sin \theta + 2) \) and then analyze the set \( S = \{a, b\} \) where \( a \) and \( b \) are the minimum and maximum values of \( y \) respectively.

### Step 1: Analyze the function inside the cosine
We start by analyzing the expression inside the cosine:
\[ x = \sin^2 \theta + \sin \theta + 2 \]
The function \( \sin^2 \theta \) varies from 0 to 1, and \( \sin \theta \) varies from -1 to 1. Therefore, we need to find the minimum and maximum values of \( x \).

### Step 2: Find the range of \( x \)
To find the minimum and maximum of \( x \):
- The minimum value occurs when \( \sin^2 \theta \) is at its minimum (0) and \( \sin \theta \) is at its minimum (-1):
\[ x_{\text{min}} = 0 + (-1) + 2 = 1 \]
- The maximum value occurs when \( \sin^2 \theta \) is at its maximum (1) and \( \sin \theta \) is at its maximum (1):
\[ x_{\text{max}} = 1 + 1 + 2 = 4 \]
Thus, the range of \( x \) is \( [1, 4] \).

### Step 3: Find the range of \( y \)
Next, we need to find the range of \( y = 2 \cos(x) \) for \( x \) in the interval \( [1, 4] \).
- The cosine function decreases from \( \cos(1) \) to \( \cos(4) \).
- Calculate \( \cos(1) \) and \( \cos(4) \):
- \( \cos(1) \approx 0.5403 \)
- \( \cos(4) \approx -0.6536 \)

Now, we can find the range of \( y \):
\[ y_{\text{min}} = 2 \cos(4) \approx 2 \times (-0.6536) \approx -1.3072 \]
\[ y_{\text{max}} = 2 \cos(1) \approx 2 \times 0.5403 \approx 1.0806 \]

Thus, the range of \( y \) is approximately:
\[ [-1.3072, 1.0806] \]

### Step 4: Define the set \( S \)
Let \( a = -1.3072 \) and \( b = 1.0806 \). Therefore, the set \( S = \{a, b\} = \{-1.3072, 1.0806\} \).

### Step 5: Analyze the options
1. If \( x \) is an integer, the only possible integer values from the range are \( -1 \) and \( 1 \). Both are negative.
2. If \( x \) is not an integer, we need to check \( \cos^{-1}(x) \) for \( x \in S \). Since \( S \) contains values from the range of \( y \), \( \cos^{-1}(x) \) will yield valid results.
3. For \( t^t \) to be defined, \( t \) must be positive or \( t = 0 \). Since \( S \) contains negative values, this option might not hold.
4. For \( x^t \) to be real, \( x \) can be negative if \( t \) is an integer.

### Conclusion
From the analysis, we conclude that:
- The range of \( y \) is approximately \( [-1.3072, 1.0806] \).
- The set \( S \) contains values that can be integers, and thus \( x \) can be negative.