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Math Help - rational functions question

  1. #1
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    rational functions question

    can someone please help me please?

    given the function f(x) = 2x/(x-4) determine the coordinates of a point on f(x) where the slope of the tangent line equals the slope of the secant line that passes through A(5,10) and B(8,4)

    the answer is x = 5, x=8; x= 6.5

    thanks in advance =)
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  2. #2
    Super Member Bacterius's Avatar
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    First, work out the slope m of the secant line :

    m = (Y_a - Y_b) / (X_a - X_b)
    Which gives m = -2

    Now that you know the slope of the secant line, you can derivate f(x) and find a derived point of f(x) which has the same slope as the secant line.
    The derivate of f(x) is :

    f'(x) = [2(x - 4) - 2x] / (x - 4)^2
    f'(x) = (2x - 8 - 2x) / (x - 4)^2
    f'(x) = (-8) / (x -4)^2

    Now you want a slope of -2, right ? Therefore, you will want to solve the derivative like this :

    f'(x) = -2
    Which gives :
    (-8) / (x - 4)^2 = -2
    Now solve this simple equation :
    -8 = -2[(x - 4)^2]
    -8 = -2(x^2 - 8x + 16)
    -8 = -2x^2 + 16x - 32
    0 = -2x^2 + 16x - 24
    Factorize this (and by the way, 0 on right-hand-side) :
    -(2x - 12)(x - 2) = 0
    Easy solve : S = (2 ; 6)
    So, x = 2 or x = 6

    Now you know that the points that have x-coordinate 2 and 6 have a slope of -2 on the first function. Substitute back into the first function to find their y-coordinate :

    f(2) = 4 / (2 - 4) = 4 / (-2) = -2
    f(6) = 12 / (6 - 4) = 12 / 2 = 6

    Let's conclude : when x = 2 or x = 6, the tangent to this point of the function has a slope of -2, which is equal to the slope of the secant line (AB).
    So, the solutions are the points (2 ; -2) and (6 ; 6).
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