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Let f: R -> R be a function such that f(x) = x³ + x²*f'(1) + x*f''(2) + f'''(3) for all x in R. Find f(2).

1. 5
2. 10
3. 6
4. -2

Correct answer: -2

Solution

Set a=f'(1), b=f''(2), c=f'''(3). Then f(x)=x³+ax²+bx+c. Derivatives: f'(x)=3x²+2ax+b, f''(x)=6x+2a, f'''(x)=6. Equations: a=f'(1)=3+2a+b => b=-a-3. b=f''(2)=12+2a. So 12+2a=-a-3 => 3a=-15 => a=-5. b=-(-5)-3=2. c=6. f(x)=x³-5x²+2x+6. f(2)=8-20+4+6=-2.

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