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Thread: Find interval of function

  1. #1
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    Find interval of function

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  2. #2
    Newbie madmartigano's Avatar
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    Inflection points occur where $\displaystyle f\prime \prime (x) = 0$

    If $\displaystyle f\prime \prime (x) = 0$ at $\displaystyle x = -1$ and $\displaystyle x = 3$, for example,
    then your intervals are $\displaystyle ( - \infty , - 1), ( - 1,3), (3, + \infty )$

    You can determine the concavity of an interval by checking any value within that interval using $\displaystyle f\prime \prime (x)$

    For the interval $\displaystyle ( - \infty , - 1)$, you could use the number -10

    If $\displaystyle f\prime \prime (-10)$ is positive, then that interval is concave up.

    If $\displaystyle f\prime \prime (-10)$ is negative, then that interval is concave down.

    Also, if $\displaystyle f\prime \prime (x)$ is negative on both sides or positive on both sides, then there is no inflection point at x.

    $\displaystyle \begin{array}{l}
    f(x) = {x^4} - 4{x^3} \\
    f\prime (x) = 4{x^3} - 12{x^2} \\
    f\prime \prime (x) = 12{x^2} - 24x = 12x(x - 2) \\ \end{array}$
    $\displaystyle f\prime \prime (x) = 0$ @ $\displaystyle x = 0, 2 \\ $

    Can you figure it out from there?
    Last edited by madmartigano; Mar 8th 2009 at 08:24 PM.
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