If ` sin 2x cos 2x cos 4x=lambda` has a solution then `lambda` lies in the interval

Options

A. `[-1//2, 1//2]`
B. `[-1//4, 1//4]`
C. `[-1//3, 1//3]`
D. none of these

Answer

To solve the problem, we need to analyze the equation given:

\[ \sin 2x \cos 2x \cos 4x = \lambda \]

### Step 1: Use the double angle identity for sine
We know that:

\[ \sin 2\theta = 2 \sin \theta \cos \theta \]

In our case, we can rewrite \(\sin 2x\) as:

\[ \sin 2x = 2 \sin x \cos x \]

Thus, we can express \(\sin 2x \cos 2x\) as:

\[ \sin 2x \cos 2x = \frac{1}{2} \sin 4x \]

### Step 2: Substitute into the equation
Substituting this back into our equation gives:

\[ \frac{1}{2} \sin 4x \cos 4x = \lambda \]

### Step 3: Use the double angle identity again
Now, we can apply the double angle identity again to \(\sin 4x \cos 4x\):

\[ \sin 4x \cos 4x = \frac{1}{2} \sin 8x \]

So we can rewrite our equation as:

\[ \frac{1}{2} \cdot \frac{1}{2} \sin 8x = \lambda \]

This simplifies to:

\[ \frac{1}{4} \sin 8x = \lambda \]

### Step 4: Determine the range of \(\lambda\)
The sine function, \(\sin 8x\), has a range of \([-1, 1]\). Therefore, we can find the range of \(\lambda\):

\[ -1 \leq \sin 8x \leq 1 \]

Multiplying the entire inequality by \(\frac{1}{4}\) gives:

\[ -\frac{1}{4} \leq \lambda \leq \frac{1}{4} \]

### Step 5: Conclusion
Thus, \(\lambda\) lies in the interval:

\[ \left[-\frac{1}{4}, \frac{1}{4}\right] \]

However, since we are looking for the interval where \(\lambda\) has solutions, we consider the open interval:

\[ \left(-\frac{1}{4}, \frac{1}{4}\right) \]

### Final Answer
The value of \(\lambda\) lies in the interval \((-1/4, 1/4)\).

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