`"If "f(x)=(x-1)^(4)(x-2)^(3)(x-3)^(2)(x-4),` then the value of `f'''(1)+f''(2)+f'(3)+f'(4)` equals

Options

A. 0
B. 50
C. 324
D. 648

Explanation

To solve the problem, we need to find the values of \( f'''(1) \), \( f''(2) \), \( f'(3) \), and \( f'(4) \) for the function

\[ f(x) = (x-1)^4 (x-2)^3 (x-3)^2 (x-4) \]

and then sum these values.

### Step 1: Calculate \( f'''(1) \)

1. **Identify the term**: The term \( (x-1)^4 \) will contribute to \( f'''(1) \).
2. **Differentiate three times**: When we differentiate \( f(x) \) three times, the \( (x-1)^4 \) term will still be present, but the other terms will vanish when evaluated at \( x = 1 \) because they will contain factors like \( (1-2) \), \( (1-3) \), and \( (1-4) \) which are all negative and will yield zero.
3. **Evaluate**: Thus, \( f'''(1) = 0 \).

### Step 2: Calculate \( f''(2) \)

1. **Identify the term**: The term \( (x-2)^3 \) will contribute to \( f''(2) \).
2. **Differentiate two times**: When we differentiate \( f(x) \) two times, the \( (x-2)^3 \) term will still be present, but the other terms will vanish when evaluated at \( x = 2 \).
3. **Evaluate**: Thus, \( f''(2) = 0 \).

### Step 3: Calculate \( f'(3) \)

1. **Identify the term**: The term \( (x-3)^2 \) will contribute to \( f'(3) \).
2. **Differentiate once**: When we differentiate \( f(x) \) once, the \( (x-3)^2 \) term will still be present, but the other terms will vanish when evaluated at \( x = 3 \).
3. **Evaluate**: Thus, \( f'(3) = 0 \).

### Step 4: Calculate \( f'(4) \)

1. **Identify the term**: The term \( (x-4) \) will contribute to \( f'(4) \).
2. **Differentiate once**: When we differentiate \( f(x) \) once, the \( (x-4) \) term will contribute a non-zero value.
3. **Evaluate**: We need to calculate \( f'(4) \):
\[ f'(x) = 4(x-1)^3(x-2)^3(x-3)^2 + 3(x-1)^4(x-2)^2(x-3)^2 + 2(x-1)^4(x-2)^3(x-3) + (x-1)^4(x-2)^3 \]
Evaluating at \( x = 4 \):
\[ f'(4) = 4(3)^3(2)^3(1)^2 + 3(3)^4(2)^2(1)^2 + 2(3)^4(2)^3(1) + (3)^4(2)^3 \]
\[ = 4 \cdot 27 \cdot 8 \cdot 1 + 3 \cdot 81 \cdot 4 \cdot 1 + 2 \cdot 81 \cdot 8 + 81 \cdot 8 \]
\[ = 864 + 972 + 162 + 648 = 2646 \]

### Final Calculation

Now we sum the results:
\[ f'''(1) + f''(2) + f'(3) + f'(4) = 0 + 0 + 0 + 2646 = 2646 \]

### Conclusion

The final answer is:
\[ \boxed{2646} \]