If $f(x) = x^{3} - x^{2}f'(1) + xf''(2) - f'''(3)$, $x \in \mathbb{R}$, then
(1) $3f(1) + f(2) = f(3)$
(2) $f(3) - f(2) = f(1)$
(3) $2f(0) - f(1) + f(3) = f(2)$
(4) $f(1) + f(2) + f(3) = f(0)$
Step-by-Step Solution
Key Concept: Let $f'(1) = a$, $f''(2) = b$, $f'''(3) = c$. Then $f(x) = x^3 - ax^2 + bx - c$. Differentiate and solve the system $a = f'(1)$, $b = f''(2)$, $c = f'''(3)$ to find $a = 3$, $b = 6$, $c = 6$.
Let $f'(1) = a$, $f''(2) = b$, $f'''(3) = c$. Then $f(x) = x^3 - ax^2 + bx - c$. Differentiating: $f'(x) = 3x^2 - 2ax + b$, $f''(x) = 6x - 2a$, $f'''(x) = 6$. So $c = 6$, $a = 3$, $b = 6$. Thus $f(x) = x^3 - 3x^2 + 6x - 6$. Computing: $f(0) = -6$, $f(1) = -2$, $f(2) = 2$, $f(3) = 12$. Check option (3): $2f(0) - f(1) + f(3) = -12 + 2 + 12 = 2 = f(2)$. Answer: (3).
Correct Answer: 3