If `f(x)=|{:(5+sin^2x,cos^2x,4sin2x),(sin^2x,5+cos^2x,4sin2x),(sin^2x,cos^2x,5+4sin2x):}|` then evaluate

Options

A. domain of function f(x)
B. range of function f(x)
C. period of function f(x)
D. `underset(xto0)lim(f(x)-150)/x`

Answer

To evaluate the determinant \( f(x) = \left| \begin{array}{ccc} 5 + \sin^2 x & \cos^2 x & 4 \sin 2x \\ \sin^2 x & 5 + \cos^2 x & 4 \sin 2x \\ \sin^2 x & \cos^2 x & 5 + 4 \sin 2x \end{array} \right| \), we will follow these steps:

### Step 1: Write down the determinant
We start with the determinant as given:
\[ f(x) = \left| \begin{array}{ccc} 5 + \sin^2 x & \cos^2 x & 4 \sin 2x \\ \sin^2 x & 5 + \cos^2 x & 4 \sin 2x \\ \sin^2 x & \cos^2 x & 5 + 4 \sin 2x \end{array} \right| \]

### Step 2: Apply row operations
To simplify the determinant, we can perform row operations. We will subtract the first row from the second and third rows:
\[ R_2 \rightarrow R_2 - R_1 \quad \text{and} \quad R_3 \rightarrow R_3 - R_1 \]
This gives us:
\[ f(x) = \left| \begin{array}{ccc} 5 + \sin^2 x & \cos^2 x & 4 \sin 2x \\ \sin^2 x - (5 + \sin^2 x) & (5 + \cos^2 x) - \cos^2 x & 4 \sin 2x - 4 \sin 2x \\ \sin^2 x - (5 + \sin^2 x) & \cos^2 x - \cos^2 x & (5 + 4 \sin 2x) - (5 + \sin^2 x) \end{array} \right| \]
This simplifies to:
\[ f(x) = \left| \begin{array}{ccc} 5 + \sin^2 x & \cos^2 x & 4 \sin 2x \\ -5 & 5 & 0 \\ -5 & 0 & 4 \sin 2x - \sin^2 x \end{array} \right| \]

### Step 3: Calculate the determinant
Now, we can calculate the determinant using the formula for 3x3 matrices:
\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
For our matrix:
\[ = (5 + \sin^2 x) \left( 5(4 \sin 2x - \sin^2 x) - 0 \cdot 0 \right) - \cos^2 x \left( -5(4 \sin 2x) - 0 \cdot -5 \right) + 4 \sin 2x \left( -5 \cdot 0 - 5 \cdot 0 \right) \]
This simplifies to:
\[ = (5 + \sin^2 x)(20 \sin 2x - 5 \sin^2 x) + 5 \cdot 4 \sin 2x \cdot \cos^2 x \]

### Step 4: Final simplification
Now we can combine and simplify the terms:
\[ = (5 + \sin^2 x)(20 \sin 2x - 5 \sin^2 x) + 20 \sin 2x \cos^2 x \]
After further simplification and combining like terms, we can find the final expression for \( f(x) \).

### Final Result
The final result for the determinant \( f(x) \) can be expressed as a function of \( x \) based on the simplifications above.