If `f(x) = |(1+sin^(2)x,cos^(2)x,4 sin 2x),(sin^(2)x,1+cos^(2)x,4 sin 2x),(sin^(2)x,cos^(2)x,1+4 sin 2x)|` What is the maximum value of f(x)?

Options

A. A. 2
B. B. 4
C. C. 6
D. D. 8

Answer

To find the maximum value of the function \( f(x) = \left| \begin{array}{ccc} 1 + \sin^2 x & \cos^2 x & 4 \sin 2x \\ \sin^2 x & 1 + \cos^2 x & 4 \sin 2x \\ \sin^2 x & \cos^2 x & 1 + 4 \sin 2x \end{array} \right| \), we will calculate the determinant step by step.

### Step 1: Write down the determinant
We start with the determinant of the given matrix:
\[ D = \left| \begin{array}{ccc} 1 + \sin^2 x & \cos^2 x & 4 \sin 2x \\ \sin^2 x & 1 + \cos^2 x & 4 \sin 2x \\ \sin^2 x & \cos^2 x & 1 + 4 \sin 2x \end{array} \right| \]

### Step 2: Simplify the determinant
We can simplify the determinant using row operations. Let's perform the operation \( R_1 \leftarrow R_1 - R_2 \) and \( R_3 \leftarrow R_3 - R_2 \).

After performing these operations, we get:
\[ D = \left| \begin{array}{ccc} 1 + \sin^2 x - \sin^2 x & \cos^2 x - (1 + \cos^2 x) & 4 \sin 2x - 4 \sin 2x \\ \sin^2 x & 1 + \cos^2 x & 4 \sin 2x \\ \sin^2 x - \sin^2 x & \cos^2 x - (1 + \cos^2 x) & 1 + 4 \sin 2x - 4 \sin 2x \end{array} \right| \]
This simplifies to:
\[ D = \left| \begin{array}{ccc} 1 & -1 & 0 \\ \sin^2 x & 1 + \cos^2 x & 4 \sin 2x \\ 0 & -1 & 1 \end{array} \right| \]

### Step 3: Calculate the determinant
Now we can calculate the determinant by expanding along the first row:
\[ D = 1 \cdot \left| \begin{array}{cc} 1 + \cos^2 x & 4 \sin 2x \\ -1 & 1 \end{array} \right| - (-1) \cdot \left| \begin{array}{cc} \sin^2 x & 4 \sin 2x \\ 0 & 1 \end{array} \right| \]

Calculating the first determinant:
\[ \left| \begin{array}{cc} 1 + \cos^2 x & 4 \sin 2x \\ -1 & 1 \end{array} \right| = (1 + \cos^2 x) \cdot 1 - (-1) \cdot 4 \sin 2x = 1 + \cos^2 x + 4 \sin 2x \]

Calculating the second determinant:
\[ \left| \begin{array}{cc} \sin^2 x & 4 \sin 2x \\ 0 & 1 \end{array} \right| = \sin^2 x \cdot 1 - 0 \cdot 4 \sin 2x = \sin^2 x \]

Putting it all together:
\[ D = 1 + \cos^2 x + 4 \sin 2x + \sin^2 x \]
Using the identity \( \cos^2 x + \sin^2 x = 1 \):
\[ D = 1 + 1 + 4 \sin 2x = 2 + 4 \sin 2x \]

### Step 4: Find the maximum value
The maximum value of \( D \) occurs when \( \sin 2x \) is maximized. The maximum value of \( \sin 2x \) is 1. Therefore:
\[ D_{\text{max}} = 2 + 4 \cdot 1 = 6 \]

### Conclusion
Thus, the maximum value of \( f(x) \) is:
\[ \boxed{6} \]