If `f(x)=|[x-2, (x-1)^2, x^3] , [(x-1), x^2, (x+1)^3] , [x,(x+1)^2, (x+2)^3]|` then coefficient of `x` in `f(x)` is

Options

A. `-4`
B. `-2`
C. `-6`
D. `0`

Explanation

To solve the problem, we need to evaluate the determinant given in the function \( f(x) \) and find the coefficient of \( x \) in the resulting polynomial.

The function is defined as:
\[ f(x) = \left| \begin{array}{ccc} x - 2 & (x - 1)^2 & x^3 \\ (x - 1) & x^2 & (x + 1)^3 \\ x & (x + 1)^2 & (x + 2)^3 \end{array} \right| \]

### Step 1: Expand the Determinant
We will expand the determinant using the first row.

\[ f(x) = (x - 2) \cdot \left| \begin{array}{cc} x^2 & (x + 1)^3 \\ (x + 1)^2 & (x + 2)^3 \end{array} \right| - (x - 1)^2 \cdot \left| \begin{array}{cc} (x - 1) & (x + 1)^3 \\ x & (x + 2)^3 \end{array} \right| + x^3 \cdot \left| \begin{array}{cc} (x - 1) & x^2 \\ x & (x + 1)^2 \end{array} \right| \]

### Step 2: Compute the 2x2 Determinants
Now we will compute each of the 2x2 determinants.

1. For the first determinant:
\[ \left| \begin{array}{cc} x^2 & (x + 1)^3 \\ (x + 1)^2 & (x + 2)^3 \end{array} \right| = x^2(x + 2)^3 - (x + 1)^3(x + 1)^2 \]

2. For the second determinant:
\[ \left| \begin{array}{cc} (x - 1) & (x + 1)^3 \\ x & (x + 2)^3 \end{array} \right| = (x - 1)(x + 2)^3 - (x + 1)^3 x \]

3. For the third determinant:
\[ \left| \begin{array}{cc} (x - 1) & x^2 \\ x & (x + 1)^2 \end{array} \right| = (x - 1)(x + 1)^2 - x^2 x \]

### Step 3: Substitute Back and Simplify
Substituting these determinants back into the expression for \( f(x) \) will yield a polynomial in \( x \).

### Step 4: Collect Like Terms
After substituting and simplifying, we will collect the terms of the polynomial and identify the coefficient of \( x \).

### Step 5: Identify the Coefficient of \( x \)
The coefficient of \( x \) can be found by looking at the simplified polynomial expression \( f(x) \).

### Final Result
After performing all calculations and simplifications, we find that the coefficient of \( x \) in \( f(x) \) is \( -2 \).

### Conclusion
Thus, the coefficient of \( x \) in \( f(x) \) is:
\[ \boxed{-2} \]