If `f(x), g(x), h(x)` are polynomials of three degree, then `phi(x)=|(f'(x),g'(x),h'(x)), (f''(x),g''(x),h''(x)), (f'''(x),g'''(x),h'''(x))|` is a polynomial of degree (where `f^n (x)` represents nth derivative of f(x))

Options

A. 3
B. 4
C. 5
D. none of these

Answer

To solve the problem, we need to analyze the polynomial functions \( f(x), g(x), h(x) \) and their derivatives.

### Step-by-Step Solution:

1. **Identify the Degree of the Polynomials**:
- Given that \( f(x), g(x), h(x) \) are polynomials of degree 3, we can express them as:
\[ f(x) = a_3 x^3 + a_2 x^2 + a_1 x + a_0 \]
\[ g(x) = b_3 x^3 + b_2 x^2 + b_1 x + b_0 \]
\[ h(x) = c_3 x^3 + c_2 x^2 + c_1 x + c_0 \]

2. **Calculate the Derivatives**:
- The first derivative of \( f(x) \):
\[ f'(x) = 3a_3 x^2 + 2a_2 x + a_1 \]
- The second derivative of \( f(x) \):
\[ f''(x) = 6a_3 x + 2a_2 \]
- The third derivative of \( f(x) \):
\[ f'''(x) = 6a_3 \]
- Similarly, we can compute the derivatives for \( g(x) \) and \( h(x) \).

3. **Construct the Determinant**:
- We need to evaluate the determinant:
\[ \phi(x) = \begin{vmatrix} f'(x) & g'(x) & h'(x) \\ f''(x) & g''(x) & h''(x) \\ f'''(x) & g'''(x) & h'''(x) \end{vmatrix} \]

4. **Determine the Degrees of Each Row**:
- The first row \( (f'(x), g'(x), h'(x)) \) consists of polynomials of degree 2.
- The second row \( (f''(x), g''(x), h''(x)) \) consists of polynomials of degree 1.
- The third row \( (f'''(x), g'''(x), h'''(x)) \) consists of constant polynomials (degree 0).

5. **Calculate the Degree of the Determinant**:
- The degree of the determinant can be found using the formula:
\[ \text{Degree}(\phi(x)) = \text{Degree of first row} + \text{Degree of second row} + \text{Degree of third row} - \text{Number of rows} + 1 \]
- Plugging in the values:
\[ \text{Degree}(\phi(x)) = 2 + 1 + 0 - 3 + 1 = 1 \]

6. **Conclusion**:
- Therefore, \( \phi(x) \) is a polynomial of degree 1.

### Final Answer:
The degree of the polynomial \( \phi(x) \) is **1**.