Figure 1 shows a sketch of the curve C with equation y = f(x), where f(x) = (x+1)(x-2)(x-4). The curve C crosses the x-axis at x=-1, x=2 and x=4. There is a minimum point at x=α and a maximum point at x=β.

Mark scheme

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

1. Part (a) B1: f(0) = (1)(-2)(-4) = 8.
2. Part (b) M1: Expands f(x) to at least 3 terms correct. E.g. (x+1)(x²-6x+8).
3. Part (b) A1: Correct terms for x³, x², x with correct signs and coefficients. x³-5x²+2x. (The constant is not required for this A1, but the derivative must be correct for the final A1).
4. Part (b) A1: Correct f'(x) = 3x²-10x+2.
5. Part (c) M1: Sets f'(x)=0 and attempts to solve a 3 term quadratic (their quadratic) using the formula or completing the square. (Requires correct substitution into formula). E.g. 3x²-10x+2=0.
6. Part (c) A1: Correct application of quadratic formula with correct values for a,b,c. Correct or unsimplified (10 ± √(100-4(3)(2)))/(2(3)) = (10 ± √76)/6.
7. Part (c) A1: For correctly simplifying the surd, e.g. (10 ± 2√19)/6.
8. Part (c) A1: For α = (5-√19)/3 and β = (5+√19)/3 (must be correctly assigned).
9. Part (d) M1: For reflection of all or part of the curve below the x-axis in the x-axis. Ignore points.
10. Part (d) B1: For curve crossing/touching x-axis at $(-1,0), (2,0), (4,0)$ and y-axis at $(0,8)$. These points can be identified on the axes.
11. Part (d) A1: A fully correct sketch with all points labelled correctly.