Let `f(theta)=(1+sin^(2)theta)(2-sin^(2)theta)`. Then for all values of `theta`

Options

A. `f(theta)gt(9)/(4)`
B. `f(theta)lt2`
C. `f(theta)gt(11)/(4)`
D. `2lef(theta)le(9)/(4)`

Answer

To solve the problem, we need to analyze the function \( f(\theta) = (1 + \sin^2 \theta)(2 - \sin^2 \theta) \).

### Step-by-Step Solution:

1. **Expand the Function**:
\[ f(\theta) = (1 + \sin^2 \theta)(2 - \sin^2 \theta) \]
Expanding this, we get:
\[ f(\theta) = 1 \cdot 2 + 1 \cdot (-\sin^2 \theta) + \sin^2 \theta \cdot 2 - \sin^4 \theta \]
\[ = 2 - \sin^2 \theta + 2\sin^2 \theta - \sin^4 \theta \]
\[ = 2 + \sin^2 \theta - \sin^4 \theta \]

2. **Rewrite the Function**:
We can rewrite \( \sin^4 \theta \) as \( (\sin^2 \theta)^2 \):
\[ f(\theta) = 2 + \sin^2 \theta (1 - \sin^2 \theta) \]
Let \( x = \sin^2 \theta \). Then, the function becomes:
\[ f(\theta) = 2 + x(1 - x) \]

3. **Find the Range of \( x \)**:
Since \( \sin^2 \theta \) varies between 0 and 1, we have:
\[ 0 \leq x \leq 1 \]

4. **Analyze the Expression \( x(1 - x) \)**:
The expression \( x(1 - x) \) is a quadratic function that opens downwards. Its maximum occurs at \( x = \frac{1}{2} \):
\[ x(1 - x) = \frac{1}{2} \left(1 - \frac{1}{2}\right) = \frac{1}{4} \]

5. **Calculate the Minimum and Maximum Values of \( f(\theta) \)**:
- **Minimum Value**: When \( x = 0 \):
\[ f(\theta) = 2 + 0 = 2 \]
- **Maximum Value**: When \( x = \frac{1}{2} \):
\[ f(\theta) = 2 + \frac{1}{4} = \frac{9}{4} \]

6. **Conclusion**:
Thus, the function \( f(\theta) \) lies between 2 and \( \frac{9}{4} \):
\[ 2 \leq f(\theta) \leq \frac{9}{4} \]

### Final Answer:
The correct option is that \( f(\theta) \) lies between 2 and \( \frac{9}{4} \).