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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?

  • pre-calculus -

    using the quadratic formula, the discriminant is

    (a^2+2a)^2 - 8a^3
    = a^2((a+2)^2 - 8a)
    = a^2(a-2)^2

    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.

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