The equation `8 cos^(4) x/2 sin^(2)x/2= x^(2) +1/x^(2), x int (0,4pi]` holds for

Options

A. No value of `x`
B. Exactly two values of x, both greater than `pi`.
C. Exactly two values of x, one smaller than `pi` and the other greater than `pi`.
D. Just one value of x

Answer

To solve the equation \(8 \cos^4 \frac{x}{2} \sin^2 \frac{x}{2} = x^2 + \frac{1}{x^2}\) for \(x \in (0, 4\pi]\), we will follow these steps:

### Step 1: Rewrite the left-hand side
The left-hand side of the equation can be rewritten using the identity for sine and cosine. We know that:
\[ \sin^2 \frac{x}{2} = 1 - \cos^2 \frac{x}{2} \]
Thus, we can express the left-hand side as:
\[ 8 \cos^4 \frac{x}{2} \sin^2 \frac{x}{2} = 8 \cos^4 \frac{x}{2} (1 - \cos^2 \frac{x}{2}) = 8 \cos^4 \frac{x}{2} - 8 \cos^6 \frac{x}{2} \]

### Step 2: Set up the inequality
The right-hand side can be analyzed using the AM-GM inequality, which states that:
\[ \frac{x^2 + \frac{1}{x^2}}{2} \geq 1 \implies x^2 + \frac{1}{x^2} \geq 2 \]
This means that we can assert:
\[ 8 \cos^4 \frac{x}{2} \sin^2 \frac{x}{2} \geq 2 \]

### Step 3: Find the critical points
To find the critical points, we can differentiate the function \(f(t) = t^3 - t^2 - t - 1\) where \(t = \cos \frac{x}{2}\). The derivative is:
\[ f'(t) = 3t^2 - 2t - 1 \]
Setting \(f'(t) = 0\) gives:
\[ 3t^2 - 2t - 1 = 0 \]
Using the quadratic formula:
\[ t = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 3 \cdot (-1)}}{2 \cdot 3} = \frac{2 \pm \sqrt{4 + 12}}{6} = \frac{2 \pm 4}{6} \]
This gives us the roots:
\[ t = 1 \quad \text{and} \quad t = -\frac{1}{3} \]

### Step 4: Analyze the intervals
We need to analyze the intervals defined by these roots. The critical points divide the number line into intervals where we can test the sign of \(f(t)\):
1. \(t < -\frac{1}{3}\)
2. \(-\frac{1}{3} < t < 1\)
3. \(t > 1\)

### Step 5: Solve for \(x\)
Now we substitute back \(t = \cos \frac{x}{2}\) and find the corresponding values of \(x\) within the interval \( (0, 4\pi] \).

### Step 6: Check integer solutions
Since \(x\) must be an integer, we check integer values of \(x\) in the interval \( (0, 4\pi] \approx (0, 12.57]\). The integer values are \(1, 2, 3, \ldots, 12\).

### Step 7: Verify solutions
We can substitute these integer values back into the original equation to verify if they satisfy it.

### Final Answer
The integer values of \(x\) that satisfy the equation are \(x = 2, 4, 6, 8, 10, 12\).