If the equation `cos^(4)theta+sin^(4) theta+lambda=0` has real solutions for `theta`, then `lambda` lies in the interval :

Options

A. `(-1/2, -1/4]`
B. `[-1, - 1/2]`
C. `[-3/2, - 5/4]`
D. `(-5/4, -1)`

Answer

To solve the equation \( \cos^4 \theta + \sin^4 \theta + \lambda = 0 \) for real solutions in terms of \( \lambda \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \cos^4 \theta + \sin^4 \theta + \lambda = 0 \]
We can use the identity \( \cos^4 \theta + \sin^4 \theta = (\cos^2 \theta + \sin^2 \theta)^2 - 2 \cos^2 \theta \sin^2 \theta \). Since \( \cos^2 \theta + \sin^2 \theta = 1 \), we can rewrite the equation as:
\[ 1 - 2 \cos^2 \theta \sin^2 \theta + \lambda = 0 \]

### Step 2: Substitute \( \cos^2 \theta \sin^2 \theta \)
Let \( x = \cos^2 \theta \sin^2 \theta \). We know that \( x = \frac{1}{4} \sin^2 2\theta \), which varies from \( 0 \) to \( \frac{1}{4} \). Thus, we can rewrite the equation as:
\[ 1 - 2x + \lambda = 0 \]
This simplifies to:
\[ \lambda = 2x - 1 \]

### Step 3: Determine the range of \( \lambda \)
Since \( x \) varies from \( 0 \) to \( \frac{1}{4} \), we can find the corresponding values of \( \lambda \):
- When \( x = 0 \):
\[ \lambda = 2(0) - 1 = -1 \]
- When \( x = \frac{1}{4} \):
\[ \lambda = 2\left(\frac{1}{4}\right) - 1 = \frac{1}{2} - 1 = -\frac{1}{2} \]

### Step 4: Conclusion
Thus, the values of \( \lambda \) for which the equation has real solutions are in the interval:
\[ \lambda \in [-1, -\frac{1}{2}] \]

### Final Answer:
The interval for \( \lambda \) is:
\[ [-1, -\frac{1}{2}] \]