Let f(x) = ax(1 - x), where a is a real number and 0 ≤ x ≤ 1. Recall that the fixed point of a function is a value of x such that f(x) = x. 

a. Without using a calculator, find the values of a, with 0 < a ≤ 4, such that f has a fixed point. Give the fixed point in terms of a.

b. Consider the polynomial g(x) = f(f(x)). Write g in terms of a and powers of x. What is its degree?

c. Graph g for a = 2, 3, and 4.

d. Find the number and location of the fixed points of g for a = 2, 3, and 4 on the interval 0 ≤ x ≤ 1.