The parabola y = x^2 and the line y = mx +b intersect at the points A (α, α^2) and B (β, β^2) as shown.

a) Explain why αβ = - b

b) Show that AB = √ (m^2 + 4b) (m^2 + 1)

c) The point P(x, x^2) lies on the parabola between A and B. Show that the area of triangle APB is given by:

Area of Triangle APB = ½ (b + mx – x^2) √ (m^2 + 4b)

d) Find the position of point P that maximizes the area of triangle APB in terms of m.

Could someone please help me solve this problem?