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Math Help - Concavity Question

  1. #1
    Member
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    Jan 2006
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    118

    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
    Posts
    319
    By the continuity of derivatives and the Intermediate Value Theorem, we know that if f(x_1) is increasing and f(x_2) is decreasing, that is, if

    f'(x_1) > 0 and f'(x_2)<0,

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

    f'(c)=0.

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

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

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