posted by cassie .
let f(x)=x^2 -(a^2 + 2a)x + 2a^3, where 0<a<2. for which value of a will the distance between the x-ints. of f be a maximum?
using the quadratic formula, the discriminant is
(a^2+2a)^2 - 8a^3
= a^2((a+2)^2 - 8a)
So, we see that
x = ((a^2+2a) ± a(a-2))/2
= a^2 or 2a
so, if the roots are a^2 and 2a, the separation is 2a-a^2. When is that a maximum? When a=1.