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y=x^3=cx and y=ax^2+bx-1 have a common tangent at (-1,-4). find a, b, andc

  1. Reiny

    I will assume the first equation is
    y = x^3 + cx (since the + and + are just a "shift" away)

    since (-1,-4) is a common point
    -4 = -1 - c
    c = 3 , and

    -4 = a(1) -b - 1
    a-b = -3 (#1)

    If they have a common tangent, then their slopes must be the same.
    3x^2 + c = 2ax + b , but c = 3
    3x^2 + 3 = 2ax + b
    again (-1,-4) must satisfy.
    3 + 3 = -2a + b
    2a - b = -6 (#2)

    #2 - #1 ---> a = -3
    back in #1
    -3 - b = -3
    b = 0

    a= -3, b=0, and c=3

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