Suppose `a,b,c,gt 1` and
`f(x)=|(a^(-x),a^(x),x),(b^(-3x),b^(3x),3x^(3)),(c^(-5x),c^(5x),5x^(5))|, x epsilon R` then f is

Options

A. a constant function
B. a polynomial of degree 5
C. an odd function
D. an even funtion

Answer

To solve the problem, we need to evaluate the determinant given by the function \( f(x) \):

\[ f(x) = \begin{vmatrix} a^{-x} & a^{x} & x \\ b^{-3x} & b^{3x} & 3x^{3} \\ c^{-5x} & c^{5x} & 5x^{5} \end{vmatrix} \]

### Step 1: Calculate the Determinant
We will calculate the determinant using the formula for a 3x3 matrix:

\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix:

- \( a = a^{-x} \), \( b = a^{x} \), \( c = x \)
- \( d = b^{-3x} \), \( e = b^{3x} \), \( f = 3x^{3} \)
- \( g = c^{-5x} \), \( h = c^{5x} \), \( i = 5x^{5} \)

Thus, we can express the determinant as:

\[ f(x) = a^{-x} \left( b^{3x} \cdot 5x^{5} - 3x^{3} \cdot c^{5x} \right) - a^{x} \left( b^{-3x} \cdot 5x^{5} - 3x^{3} \cdot c^{-5x} \right) + x \left( b^{-3x} \cdot c^{5x} - b^{3x} \cdot c^{-5x} \right) \]

### Step 2: Simplifying the Determinant
Now we will simplify each term in the determinant:

1. **First Term:**
\[ a^{-x} \left( b^{3x} \cdot 5x^{5} - 3x^{3} \cdot c^{5x} \right) \]

2. **Second Term:**
\[ - a^{x} \left( b^{-3x} \cdot 5x^{5} - 3x^{3} \cdot c^{-5x} \right) \]

3. **Third Term:**
\[ + x \left( b^{-3x} \cdot c^{5x} - b^{3x} \cdot c^{-5x} \right) \]

### Step 3: Combine and Factor
Now we can combine the terms and factor out common elements where possible. The expression becomes:

\[ f(x) = a^{-x} b^{3x} \cdot 5x^{5} - a^{-x} 3x^{3} c^{5x} - a^{x} b^{-3x} \cdot 5x^{5} + a^{x} 3x^{3} c^{-5x} + x (b^{-3x} c^{5x} - b^{3x} c^{-5x}) \]

### Step 4: Determine the Nature of \( f(x) \)
Now we need to analyze \( f(x) \) to determine its properties:

1. **Check if \( f(x) \) is a constant function:**
Since \( f(x) \) contains terms with \( x \) raised to various powers, it is not a constant function.

2. **Check if \( f(x) \) is a polynomial function:**
The highest degree term in \( f(x) \) is \( 5x^{5} \), indicating that \( f(x) \) is indeed a polynomial of degree 5.

3. **Check if \( f(x) \) is even or odd:**
To check if \( f(x) \) is even or odd, we compute \( f(-x) \) and compare it to \( f(x) \):
- If \( f(-x) = f(x) \), then \( f(x) \) is even.
- If \( f(-x) = -f(x) \), then \( f(x) \) is odd.

After performing the calculations, we find that \( f(-x) = f(x) \), confirming that \( f(x) \) is an even function.

### Final Conclusion
Thus, we conclude that:

- \( f(x) \) is a polynomial of degree 5.
- \( f(x) \) is an even function.