Consider the equation below. (If an answer does not exist, enter DNE.) f(x) = x^3 − 3x^2 − 9x + 4 (a) Find the interval on which f is increa

Question

Consider the equation below. (If an answer does not exist, enter DNE.) f(x) = x^3 − 3x^2 − 9x + 4 (a) Find the interval on which f is increasing. (Enter your answer using interval notation.)

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Emery 5 days 2021-09-13T16:26:25+00:00 1 Answer 0

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    2021-09-13T16:27:40+00:00

    Answer:

    (-∞, -1 )∪(3, ∞)

    Step-by-step explanation:

    Given function,

    f(x) = x^3 - 3x^2 - 9x + 4

    Differentiating with respect to x,

    f'(x)=3x^2-6x-9

    For increasing or decreasing,

    f'(x) = 0

    \implies 3x^2-6x-9=0

    By quadratic formula,

    x=\frac{-(-6)\pm \sqrt{(-6)^2-4\times 3\times -9}}{6}

    =\frac{ 6\pm \sqrt{36+108}}{6}

    =\frac{6\pm \sqrt{144}}{6}

    \frac{6\pm 12}{6}

    \implies x = 3\text{ or } x = -1

    In interval (-∞, -1 ), f'(x) = positive,

    f(x) is increasing on (-∞, -1 ),

    In interval (-1, 3), f'(x) = negative,

    f(x) is decreasing on (-1, 3),

    In interval (3, ∞), f'(x) = positive,

    f(x) is increasing on (3, ∞),

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