If `f (x )= (x-1) ^(4) (x-2) ^(3) (x-3) ^(2)` then the value of `f '(1) +f''(2) +f''(3)` is:

Options

A. 0
B. 1
C. 2
D. 6

Answer

To solve the problem, we need to find \( f'(1) + f''(2) + f''(3) \) for the function \( f(x) = (x-1)^4 (x-2)^3 (x-3)^2 \).

### Step 1: Find \( f'(x) \)

We will use the product rule to differentiate \( f(x) \). The product rule states that if \( f(x) = g(x) h(x) \), then \( f'(x) = g'(x) h(x) + g(x) h'(x) \).

Let:
- \( g(x) = (x-1)^4 \)
- \( h(x) = (x-2)^3 (x-3)^2 \)

Now, we need to differentiate \( g(x) \) and \( h(x) \).

1. **Differentiate \( g(x) \)**:
\[ g'(x) = 4(x-1)^3 \]

2. **Differentiate \( h(x) \)** using the product rule:
Let \( h_1(x) = (x-2)^3 \) and \( h_2(x) = (x-3)^2 \).
- \( h_1'(x) = 3(x-2)^2 \)
- \( h_2'(x) = 2(x-3) \)

Now apply the product rule:
\[ h'(x) = h_1'(x) h_2(x) + h_1(x) h_2'(x) = 3(x-2)^2 (x-3)^2 + (x-2)^3 \cdot 2(x-3) \]

Now we can combine these to find \( f'(x) \):
\[ f'(x) = g'(x) h(x) + g(x) h'(x) \]

### Step 2: Evaluate \( f'(1) \)

Substituting \( x = 1 \):
- \( g(1) = (1-1)^4 = 0 \)
- \( h(1) = (1-2)^3 (1-3)^2 = (-1)^3 \cdot (-2)^2 = -1 \cdot 4 = -4 \)

Thus:
\[ f'(1) = g'(1) h(1) + g(1) h'(1) = 4(0) + 0 = 0 \]

### Step 3: Find \( f''(x) \)

To find \( f''(x) \), we differentiate \( f'(x) \) again. However, we can directly evaluate \( f''(2) \) and \( f''(3) \) without finding the entire second derivative.

### Step 4: Evaluate \( f''(2) \)

At \( x = 2 \):
- \( g(2) = (2-1)^4 = 1 \)
- \( h(2) = (2-2)^3 (2-3)^2 = 0 \)

Thus:
\[ f'(2) = g'(2) h(2) + g(2) h'(2) = 4(1) \cdot 0 + 1 \cdot h'(2) = h'(2) \]

Since \( h(2) = 0 \), \( f'(2) = 0 \).

Now, for \( f''(2) \), we note that since \( f'(2) = 0 \) and \( h(2) = 0 \), we can conclude \( f''(2) = 0 \).

### Step 5: Evaluate \( f''(3) \)

At \( x = 3 \):
- \( g(3) = (3-1)^4 = 16 \)
- \( h(3) = (3-2)^3 (3-3)^2 = 0 \)

Thus:
\[ f'(3) = g'(3) h(3) + g(3) h'(3) = g(3) \cdot h'(3) \]

Since \( h(3) = 0 \), \( f'(3) = 0 \).

Thus, \( f''(3) = 0 \).

### Step 6: Combine the results

Now we can sum the results:
\[ f'(1) + f''(2) + f''(3) = 0 + 0 + 0 = 0 \]

### Final Answer
The value of \( f'(1) + f''(2) + f''(3) \) is \( \boxed{0} \).