A curve has equation y = x^3 - 6x^2 + 9x + 2.

Mark scheme

Marking points from the official mark scheme. The mark or mark type is shown in brackets.

1. "(a)"
2. M1 for differentiating at least two terms correctly.
3. A1 for dy/dx = 3x^2 - 12x + 9 (accept unsimplified forms, e.g., 3x^2 - 12x^1 + 9x^0).
4. "(b)"
5. M1 for setting dy/dx = 0.
6. M1 for solving their quadratic equation for dy/dx = 0. (factorising, formula or calculator).
7. A1 for x = 1 and x = 3 (both correct).
8. A1 for (1, 6) and (3, 2) (both correct pairs, requires substitution into original y equation).
9. "(c)"
10. M1 for finding the second derivative, d^2y/dx^2 = 6x - 12.
11. A1 for correctly evaluating d^2y/dx^2 for both x-values: d^2y/dx^2 = -6 at x=1 and d^2y/dx^2 = 6 at x=3 (or for finding values of d^2y/dx^2).
12. A1 for correctly stating nature: (1, 6) is a maximum point and (3, 2) is a minimum point (must link to coordinates or x-values).