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How many integer values of a are there such that
has no local extrema?

  • calculus -

    Assuming you meant


    we want a derivative with no zeros. So,

    f' = 3x^2 + 2ax + 8a
    f" = 6x+2a

    this will have no zeros if the discriminant is negative, so we need

    (2a)^2 - 4(3)(8a) < 0
    4a^2 - 96a < 0
    4a(a-24) < 0

    So 0<a<24

    At a=0 and a=24, f"(0) when f'=0, so there's an inflection point, so no extrema.

    So, 0 <= a <= 24

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