Let `f(x)=|(a^2+(b^2+c^2)cosx,ab(1-cosx),ac(1-cosx)),(ba(1-cosx),b^2+(c^2+a^2)cosx,bc(1-cosx)),(ca(1-cosx),cb(1-cosx),c^2+(a^2+b^2)cosx)|` where `"x"in[-pi/2,pi/2]` and a,b,c
Max value of f (x) =

Options

A. `sum a^2`
B. `(suma^2)^2`
C. `(suma^2)^3`
D. None

Answer

To find the maximum value of the function \( f(x) \) defined as the determinant of the matrix:

\[ f(x) = \begin{vmatrix} a^2 + (b^2 + c^2) \cos x & ab(1 - \cos x) & ac(1 - \cos x) \\ ba(1 - \cos x) & b^2 + (c^2 + a^2) \cos x & bc(1 - \cos x) \\ ca(1 - \cos x) & cb(1 - \cos x) & c^2 + (a^2 + b^2) \cos x \end{vmatrix} \]

we will simplify the determinant and find its maximum value.

### Step 1: Factor out common terms
We can factor out \( abc \) from the determinant. This can be done by multiplying the first column by \( b \), the second column by \( c \), and the third column by \( a \).

\[ f(x) = \frac{1}{abc} \begin{vmatrix} a^3 + a(b^2 + c^2) \cos x & ab^2(1 - \cos x) & a^2c(1 - \cos x) \\ b^2a(1 - \cos x) & b^3 + b(c^2 + a^2) \cos x & b^2c(1 - \cos x) \\ c^2a(1 - \cos x) & c^2b(1 - \cos x) & c^3 + c(a^2 + b^2) \cos x \end{vmatrix} \]

### Step 2: Rewrite the determinant
Now we can rewrite the determinant in a more manageable form. We can denote \( A = a^2 + b^2 + c^2 \) and simplify the determinant.

\[ f(x) = \frac{1}{abc} \begin{vmatrix} A \cos x & ab(1 - \cos x) & ac(1 - \cos x) \\ ba(1 - \cos x) & A \cos x & bc(1 - \cos x) \\ ca(1 - \cos x) & cb(1 - \cos x) & A \cos x \end{vmatrix} \]

### Step 3: Calculate the determinant
Using properties of determinants, we can calculate the determinant. The structure of the matrix suggests that we can use cofactor expansion or other determinant properties to simplify it.

After performing the calculations (which may involve some algebraic manipulation), we find that:

\[ f(x) = \frac{(A \cos x)^3 - 3(A \cos x)(ab(1 - \cos x))(ac(1 - \cos x))}{abc} \]

### Step 4: Maximize the function
To find the maximum value of \( f(x) \), we need to analyze the expression we obtained. The maximum value occurs when \( \cos x \) is at its maximum, which is 1, leading us to evaluate:

\[ f(0) = \frac{(A)^3}{abc} \]

### Step 5: Conclusion
Thus, the maximum value of \( f(x) \) is:

\[ \text{Max value of } f(x) = \frac{(a^2 + b^2 + c^2)^3}{abc} \]