Consider the cubic polynomial f(x) = x(x - a)(x - b), where 0 ≤ a ≤ b.

a. For a fixed value of b, find the functionFor what value of a (which depends on b) is F(a) = 0?

b. For a fixed value of b, find the function A(a) that gives the area of the region bounded by the graph of f and the x-axis between x = 0 and x = b. Graph this function and show that it has a minimum at a = b/2. What is the maximum value of A(a), and where does it occur (in terms of b)?

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F(a) = Sf(x) dx.