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    Differentiation : Existence of Solutions

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    I want to prove, for the numbers a and b, that the following equation has exactly three solutions if and only if 4a^3 + 27b^2 < 0:

    x^3 + ax + b = 0,

    x in R

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    Solution Preview

    Because, f(x) = x^3 + ax + b

    => f'(x) = 3x^2 + a

    for maxima-minima:

    f'(x) = 0 => x = +sqrt(-a/3) or x = -sqrt(-a/3)

    f''(x) = 6x

    f''(x=+sqrt(-a/3)) = ...

    Solution Summary

    It is proved that there are a set number of solutions for a particular function. Maxima and minima's are analyzed for functions.