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Thread: Decreasing function

  1. #1
    Member looi76's Avatar
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    Unhappy Decreasing function

    Question:

    The diagram shows the curve $\displaystyle y = x^3 - 3x^2 - 9x + k$, where k is constant. The curve has a minimum point on the x-axis.
    (i) Find the value of $\displaystyle k$.
    (ii) Find the coordinates of the maximum point of the curve.
    (iii) State the set of values of $\displaystyle x$ for which $\displaystyle x^3 - 3x^2 - 9x + k$ is a decreasing function of $\displaystyle x$.

    Attempt:
    (i) $\displaystyle k = 27$
    (ii) $\displaystyle (-1,32)$
    (iii) I don't know how to get the decreasing order?!...
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  2. #2
    GAMMA Mathematics
    colby2152's Avatar
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    Keep in mind that the function is decreasing before it hits that minimum.
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  3. #3
    Member looi76's Avatar
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    Quote Originally Posted by colby2152 View Post
    Keep in mind that the function is decreasing before it hits that minimum.
    Thanks colby2152, but I don't know which formula to use!..
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  4. #4
    Lord of certain Rings
    Isomorphism's Avatar
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    Quote Originally Posted by looi76 View Post
    Question:

    The diagram shows the curve $\displaystyle y = x^3 - 3x^2 - 9x + k$, where k is constant. The curve has a minimum point on the x-axis.
    (i) Find the value of $\displaystyle k$.
    (ii) Find the coordinates of the maximum point of the curve.
    (iii) State the set of values of $\displaystyle x$ for which $\displaystyle x^3 - 3x^2 - 9x + k$ is a decreasing function of $\displaystyle x$.

    Attempt:
    (i) $\displaystyle k = 27$
    (ii) $\displaystyle (-1,32)$
    (iii) I don't know how to get the decreasing order?!...
    Well actually its quite clear from the graph that it is decreasing between the maximum and the minimum. So for -1 < x < 3, we have the function decreasing....

    A calculus way of showing it is computing the derivative of f(x) and if the derivative is less than 0 then the function is decreasing.

    So $\displaystyle f'(x) = 3x^2 - 6x -9 < 0 \Rightarrow 3(x^2 - 2x - 3) = 3(x-3)(x+1) < 0 \Rightarrow x \in (-1,3)$
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