If sin x cos x cos 2x = `lambda` has a solution, then `lambda` lies in the interval

Options

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

Answer

To solve the problem, we need to determine the range of the variable \( \lambda \) in the equation \( \sin x \cos x \cos 2x = \lambda \).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ \sin x \cos x \cos 2x = \lambda \]

2. **Use the double angle identity for sine**:
Recall that \( \sin 2x = 2 \sin x \cos x \). Therefore, we can rewrite \( \sin x \cos x \) as:
\[ \sin x \cos x = \frac{1}{2} \sin 2x \]
Substituting this into the equation gives:
\[ \frac{1}{2} \sin 2x \cos 2x = \lambda \]

3. **Use the double angle identity for sine again**:
We know that \( \sin 2\theta = 2 \sin \theta \cos \theta \). Thus, we can express \( \sin 2x \cos 2x \) as:
\[ \sin 2x \cos 2x = \frac{1}{2} \sin 4x \]
Substituting this into the equation gives:
\[ \frac{1}{2} \cdot \frac{1}{2} \sin 4x = \lambda \]
Simplifying this, we have:
\[ \frac{1}{4} \sin 4x = \lambda \]

4. **Express \( \sin 4x \) in terms of \( \lambda \)**:
Rearranging the equation gives:
\[ \sin 4x = 4\lambda \]

5. **Determine the range of \( \sin 4x \)**:
The sine function has a range of \([-1, 1]\). Therefore, we can set up the inequality:
\[ -1 \leq 4\lambda \leq 1 \]

6. **Solve for \( \lambda \)**:
Dividing the entire inequality by 4 gives:
\[ -\frac{1}{4} \leq \lambda \leq \frac{1}{4} \]

7. **Write the final interval for \( \lambda \)**:
Thus, we conclude that:
\[ \lambda \in \left[-\frac{1}{4}, \frac{1}{4}\right] \]

### Final Answer:
The value of \( \lambda \) lies in the interval:
\[ \lambda \in \left[-\frac{1}{4}, \frac{1}{4}\right] \]