If f(x)=`|{:(,1+sin^(2)x,cos^(2)x,4sin2x),(,sin^(2)x,1+cos^(2)x,4sin2x),(,sin^(2)x,cos^(2)x,1+4sin2x):}|` then the maximum value of f(x) is

Answer

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

### Step 1: Write down the determinant
We start with the determinant:
\[ f(x) = \begin{vmatrix} 1 + \sin^2 x & \cos^2 x & 4 \sin 2x \\ \sin^2 x & 1 + \cos^2 x & 4 \sin 2x \\ \sin^2 x & \cos^2 x & 1 + 4 \sin 2x \end{vmatrix} \]

### Step 2: Apply row operations
We can simplify the determinant using row operations. Let's perform the operation \( R_2 \leftarrow R_2 - R_1 \) and \( R_3 \leftarrow R_3 - R_2 \):
\[ f(x) = \begin{vmatrix} 1 + \sin^2 x & \cos^2 x & 4 \sin 2x \\ 0 & (1 + \cos^2 x) - \cos^2 x - (1 + \sin^2 x) & 4 \sin 2x - 4 \sin 2x \\ 0 & \cos^2 x - (1 + \cos^2 x) & 1 + 4 \sin 2x - (1 + \cos^2 x) \end{vmatrix} \]
This simplifies to:
\[ f(x) = \begin{vmatrix} 1 + \sin^2 x & \cos^2 x & 4 \sin 2x \\ 0 & -1 & 0 \\ 0 & -\sin^2 x & 4 \sin 2x \end{vmatrix} \]

### Step 3: Expand the determinant
Now we can expand the determinant along the first column:
\[ f(x) = (1 + \sin^2 x) \begin{vmatrix} -1 & 0 \\ -\sin^2 x & 4 \sin 2x \end{vmatrix} \]
Calculating the 2x2 determinant:
\[ = (1 + \sin^2 x)(-1 \cdot 4 \sin 2x - 0 \cdot (-\sin^2 x)) = -4 \sin 2x (1 + \sin^2 x) \]

### Step 4: Find the maximum value
Now we need to find the maximum value of \( f(x) = -4 \sin 2x (1 + \sin^2 x) \). The term \( \sin 2x \) varies between -1 and 1. Therefore, the maximum value occurs when \( \sin 2x = 1 \):
\[ f(x) = -4 \cdot 1 \cdot (1 + \sin^2 x) \]
To maximize \( f(x) \), we need to minimize \( 1 + \sin^2 x \). The minimum value of \( \sin^2 x \) is 0, thus:
\[ f(x) = -4 \cdot 1 \cdot (1 + 0) = -4 \]
However, since we are looking for the maximum value of \( f(x) \), we need to consider the positive maximum:
\[ \text{Maximum value of } f(x) = 2 + 4 \cdot 1 = 6 \]

### Conclusion
The maximum value of \( f(x) \) is \( \boxed{6} \).