Let \(f'(x) = 6{x^2} - 5\). Given that \(f(2) =  - 3\), find \(f(x)\).

evidence of antidifferentiation     (M1)

eg\(\;\;\;f = \int {f'} \)

correct integration (accept absence of \(C\))     (A1)(A1)

\(f(x) = \frac{{6{x^3}}}{3} - 5x + C,{\text{ }}2{x^3} - 5x\)

attempt to substitute \((2,{\text{ }} - 3)\) into their integrated expression (must have \(C\))     M1

eg\(\;\;\;2{(2)^3} - 5(2) + C =  - 3,{\text{ }}16 - 10 + C =  - 3\)

Note:     Award M0 if substituted into original or differentiated function.

correct working to find \(C\)     (A1)

eg\(\;\;\;16 - 10 + C =  - 3,{\text{ }}6 + C =  - 3,{\text{ }}C =  - 9\)

\(f(x) = 2{x^3} - 5x - 9\)     A1     N4

[6 marks]