Figure 1 shows a sketch of the curve C with equation y = f(x) - Edexcel - A-Level Maths Pure - Question 11 - 2013 - Paper 2

To find the coefficients a, b, and c, we utilize the conditions given:

1. Since the curve passes through the point (−1, 0):
   Substituting x = -1 and y = 0 into the equation gives:

   $0 = (-1)^3 + a(-1)^2 + b(-1) + c$
   $0 = -1 + a - b + c$
   Thus, we have:
   $a - b + c = 1 ag{1}$

2. Since the curve touches the x-axis at the point (2, 0):
   Substituting x = 2 and y = 0 gives:

   $0 = (2)^3 + a(2)^2 + b(2) + c$
   $0 = 8 + 4a + 2b + c$
   Therefore:
   $4a + 2b + c = -8 ag{2}$

3. The curve has a maximum at the point (0, 4):
   Substituting x = 0 and y = 4 gives:

   $4 = (0)^3 + a(0)^2 + b(0) + c$
   Hence, we find:
   $c = 4 ag{3}$

Using (3) in (1) and (2): From (1):
$a - b + 4 = 1$
$a - b = -3 ag{4}$

From (2):
$4a + 2b + 4 = -8$

ightarrow 2a + b = -6 ag{5}$$ Now, solving equations (4) and (5): From (4): $$b = a + 3$$ Substituting into (5): $$2a + (a + 3) = -6$$ $$3a + 3 = -6$$ Thus, $$3a = -9 ightarrow a = -3$$ Substituting back to find b: $$b = -3 + 3 = 0$$ Therefore, the values of a, b, and c are: - a = -3 - b = 0 - c = 4.