Figure 1 shows a sketch of the curve C with equation y = f(x) - Edexcel - A-Level Maths Pure - Question 10 - 2013 - Paper 2

To find a, b, and c, we start with the form of the equation:

$y = x^3 + ax^2 + bx + c$

Using the given points:

1. Since the curve passes through the point (-1, 0), we substitute to find: $0 = (-1)^3 + a(-1)^2 + b(-1) + c\n 0 = -1 + a - b + c \n a - b + c = 1 \quad (1)$

2. At (2, 0), the curve touches the x-axis, meaning it is a double root: $0 = (2)^3 + a(2)^2 + b(2) + c \n 0 = 8 + 4a + 2b + c \n 4a + 2b + c = -8 \quad (2)$

3. The maximum point (0, 4) gives us: $4 = (0)^3 + a(0)^2 + b(0) + c \n c = 4 \quad (3)$

Substituting c in equations (1) and (2):

1. From (1): $a - b + 4 = 1 \Rightarrow a - b = -3 \quad (4)$

2. Substitute c into (2): $4a + 2b + 4 = -8 \Rightarrow 4a + 2b = -12 \Rightarrow 2a + b = -6 \quad (5)$

Now, solve equations (4) and (5):

• From (4): $b = a + 3$. Substitute into (5): $2a + (a + 3) = -6 \Rightarrow 3a + 3 = -6 \Rightarrow 3a = -9 \Rightarrow a = -3$
• Thus, $b = -3 + 3 = 0$.

Final values: a = -3, b = 0, c = 4.