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Thread: Concavity Question

  1. #1
    Member
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    Concavity Question

    Consider the function f(x) = 7x + 8x ^-1. For this function there are four important intervals: (-\infty, A], [A,B),(B,C), and [C,\infty) where A, and C are the critical numbers and the function is not defined at B.

    Find A
    and B
    and C

    For each of the following intervals, tell whether f(x) is increasing or decreasing.

    (i infty, A]:
    [A,B):
    (B,C]:
    [C,infty):


    Any help will be appreciated!
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  2. #2
    Senior Member
    Joined
    Dec 2008
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    By the continuity of derivatives and the Intermediate Value Theorem, we know that if $\displaystyle f(x_1)$ is increasing and $\displaystyle f(x_2)$ is decreasing, that is, if

    $\displaystyle f'(x_1) > 0$ and $\displaystyle f'(x_2)<0$,

    and if $\displaystyle f'(x)$ is defined on $\displaystyle (x_1,x_2)$,then there must be a middle value $\displaystyle c$ such that

    $\displaystyle f'(c)=0.$

    Therefore, between the values of $\displaystyle x$ at which critical points occur, $\displaystyle f(x)$ can only be increasing or decreasing, but not both. As a result, we may calculate the value of $\displaystyle f'(x)$ at 'test points' within these regions to determine the behavior of $\displaystyle f(x)$ over that region. In our case, the test points can be any points in the intervals

    $\displaystyle (-\infty,A]\;\;\;\;\;\;\;\;\;\;[A,B)\;\;\;\;\;\;\;\;\;\;(B,C]\;\;\;\;\;\;\;\;\;\;[C,\infty)$

    To find the values of $\displaystyle A$, $\displaystyle B$, and $\displaystyle C$, we calculate $\displaystyle f'(x)$. When is $\displaystyle f'(x)=0$? When is $\displaystyle f'(x)$ undefined?
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