If f'(x) = 6x^2 - 4x + 1 and f(1) = 5, find the function f(x).
Answer and explanation
Correct answer: D. f(x) = 2x^3 - 2x^2 + x + 4
- Step 1: Antidifferentiate: f(x) = 2x^3 - 2x^2 + x + C. Step 2. Use initial condition f(1) = 5: 2(1)^3 - 2(1)^2 + 1 + C = 5 => 2 - 2 + 1 + C = 5 => 1 + C = 5 => C = 4. Step 3. Thus f(x) = 2x^3 - 2x^2 + x + 4. Answer: f(x) = 2x^3 - 2x^2 + x + 4. Why not the others: C = 2 miscalculates 5 - 1; C = 5 fails to subtract f(1) offset; 12x - 4 takes another derivative.
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