Part 2a:
The general form of a cubic curve is -
y = ax^3 + bx^2 + cx + d
(0,0) (1,2) (2,1)
To find the values of a, b, c and d, we just need to substitute the values of x and y in each of the three coordinates, and build a system of 3 equations. These equations are:
(0,0)
0 = d
(1,2)
2 = a + b + c
(2,1)
1 = 8a + 4b + 2c
We also know that (1,2) is a stationary point. This tells us that the first derivative of the general equation is equal to 0 at x=1.
first derivative: 3ax^2 + 2bx + c
Therefore: 3a(1)^2 + 2b(1) + c = 0
3a + 2b + c = 0
Thus, the three equations we have are,
1. a + b + c = 2
2. 8a + 4b + 2c = 1
3. 3a + 2b + c= 0
We can multiply equation 3 by 2, and then subtract that from equation 2. This gives us,
2a = 1
a = 1/2
From this, we get the two equations:
b + c = 3/2
4b + 2c = -3
By using elimination method, the values of b and c are,
b = -3
c = 4.5
Therefore, the cubic equation we get is:
y = (1/2)x^3 -3x^2 + 4.5x