Given that \(f '(x) = 3x^{2} - 6x + 1\) and f(3) = 5, find f(x).
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Correct Answer: Option C
Explanation:
\(f ' (x) = 3x^{2} - 6x + 1\)
\(f(x) = \int (3x^{2} - 6x + 1) \mathrm {d} x\)
= \(x^{3} - 3x^{2} + x + c\)
\(f(3) = 5 = 3^{3} - 3(3^{2}) + 3 + c\)
\(27 - 27 + 3 + c = 5 \implies 3 + c = 5\)
\(c = 2\)
\(f(x) = x^{3} - 3x^{2} + x + 2\)
\(f ' (x) = 3x^{2} - 6x + 1\)
\(f(x) = \int (3x^{2} - 6x + 1) \mathrm {d} x\)
= \(x^{3} - 3x^{2} + x + c\)
\(f(3) = 5 = 3^{3} - 3(3^{2}) + 3 + c\)
\(27 - 27 + 3 + c = 5 \implies 3 + c = 5\)
\(c = 2\)
\(f(x) = x^{3} - 3x^{2} + x + 2\)