`" If " f(x) = |{:(3,,3x,,3x^(2)+2a^(2)),(3x,,3x^(2)+2a^(2),,3x^(3)+6a^(2)x),(3x^(2)+2a^(2),,3x^(3)+6a^(2)x,,3x^(4)+12a^(2)x^(2)+2a^(4)):}|` then

Options

A. f'(x)=0
B. y=f(x) is a straight line parallel to x-axis
C. `int_(0)^(2) f(x) dx=32 a^(4)`
D. none of these

Explanation

To solve the determinant given in the function \( f(x) \), we will follow a systematic approach to simplify the determinant step by step.

### Step 1: Write the Determinant
We start with the determinant defined as:

\[ f(x) = \begin{vmatrix} 3 & 3x & 3x^2 + 2a^2 \\ 3x & 3x^2 + 2a^2 & 3x^3 + 6a^2x \\ 3x^2 + 2a^2 & 3x^3 + 6a^2x & 3x^4 + 12a^2x^2 + 2a^4 \end{vmatrix} \]

### Step 2: Apply Column Operations
We will perform column operations to simplify the determinant. We can subtract \( x \) times the first column from the second column and \( x \) times the second column from the third column.

\[ C_2 \rightarrow C_2 - xC_1 \]
\[ C_3 \rightarrow C_3 - xC_2 \]

After performing these operations, the determinant becomes:

\[ f(x) = \begin{vmatrix} 3 & 0 & 3x^2 + 2a^2 - x(3x) \\ 3x & 0 & 3x^3 + 6a^2x - x(3x^2 + 2a^2) \\ 3x^2 + 2a^2 & 0 & 3x^4 + 12a^2x^2 + 2a^4 - x(3x^3 + 6a^2x) \end{vmatrix} \]

### Step 3: Simplify the Determinant
Now we simplify the entries in the determinant.

The second column becomes zero, and we can simplify the third column:

\[ 3x^2 + 2a^2 - 3x^2 = 2a^2 \]
\[ 3x^3 + 6a^2x - (3x^3 + 2a^2x) = 4a^2x \]
\[ 3x^4 + 12a^2x^2 + 2a^4 - (3x^4 + 6a^2x^2) = 6a^2x^2 + 2a^4 \]

Thus, we have:

\[ f(x) = \begin{vmatrix} 3 & 0 & 2a^2 \\ 3x & 0 & 4a^2x \\ 3x^2 + 2a^2 & 0 & 6a^2x^2 + 2a^4 \end{vmatrix} \]

### Step 4: Factor Out Common Terms
Next, we can factor out the common terms from the rows:

\[ f(x) = 3 \cdot 0 \cdot \text{determinant} = 0 \]

Since the second column is entirely zero, the determinant evaluates to zero.

### Step 5: Conclusion
Thus, we conclude that:

\[ f(x) = 0 \]