If `x=cos^2theta+sin^4theta` then for all real values of `theta`

Options

A. `x in[1,2]`
B. `x in[13//16,1]`
C. `x in[3//4,13//16]`
D. `x in[3//4,1]`

Explanation

To solve the problem, we need to find the range of the expression \( x = \cos^2 \theta + \sin^4 \theta \) for all real values of \( \theta \).

### Step-by-Step Solution:

1. **Rewrite the Expression**:
We start with the given expression:
\[ x = \cos^2 \theta + \sin^4 \theta \]
We can rewrite \( \sin^4 \theta \) in terms of \( \sin^2 \theta \):
\[ \sin^4 \theta = (\sin^2 \theta)^2 \]
Thus, we can express \( x \) as:
\[ x = \cos^2 \theta + (\sin^2 \theta)^2 \]

2. **Use the Pythagorean Identity**:
We know from the Pythagorean identity that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Let \( y = \sin^2 \theta \). Then \( \cos^2 \theta = 1 - y \). Substituting this into our expression for \( x \):
\[ x = (1 - y) + y^2 \]
Simplifying gives:
\[ x = 1 - y + y^2 \]

3. **Form a Quadratic Equation**:
The expression \( x = y^2 - y + 1 \) is a quadratic in \( y \). We can analyze this quadratic to find its minimum value. The general form of a quadratic \( ay^2 + by + c \) has its vertex at \( y = -\frac{b}{2a} \):
\[ y = -\frac{-1}{2 \cdot 1} = \frac{1}{2} \]

4. **Calculate the Value at the Vertex**:
Substitute \( y = \frac{1}{2} \) back into the expression for \( x \):
\[ x = \left(\frac{1}{2}\right)^2 - \left(\frac{1}{2}\right) + 1 = \frac{1}{4} - \frac{1}{2} + 1 = \frac{1}{4} - \frac{2}{4} + \frac{4}{4} = \frac{3}{4} \]

5. **Determine the Maximum Value**:
Since \( y = \sin^2 \theta \) can take values from \( 0 \) to \( 1 \), we need to evaluate \( x \) at these endpoints:
- When \( y = 0 \):
\[ x = 1 - 0 + 0^2 = 1 \]
- When \( y = 1 \):
\[ x = 1 - 1 + 1^2 = 1 \]

6. **Conclusion**:
The minimum value of \( x \) occurs at \( y = \frac{1}{2} \) which gives \( x = \frac{3}{4} \), and the maximum value occurs at the endpoints \( y = 0 \) or \( y = 1 \) which gives \( x = 1 \). Thus, the range of \( x \) is:
\[ \frac{3}{4} \leq x \leq 1 \]

### Final Answer:
The value of \( x \) lies between \( \frac{3}{4} \) and \( 1 \).