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    NationalMediumCalculus10–12

    Monotonicity and Stationary Points of a Cubic

    Monotonia e Pontos Estacionários de uma Função Cúbica

    Consider f(x)=x3−3x2+4f(x) = x^{3} - 3x^{2} + 4. Determine the intervals on which ff is increasing and those on which it is decreasing, and classify its stationary points.

    What decides increase and decrease is the sign of the derivative, not its size. Find where the derivative vanishes, then study the sign on each side of those points.

    Solution

    Step 1 of 6

    1. 1.f′(x)=3x2−6x=3x(x−2)f'(x) = 3x^{2} - 6x = 3x(x - 2)