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Find the values of a,b, and c if the parabola y=a(x^2)+bx+c is tangent to the line y=-2x+3 at (2,-1) and has a critical point when x=3

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  1. y' = 2ax + b

    we want y'(3) = 0, so
    6a+b = 0

    the line -2x+3 has slope -2, so
    we want y'(2) = -2, so
    4a+b = -2

    and we have (a,b) = (1,-6)

    y = x^2 - 6x + c

    We know that y(2) = -1, so

    -1 = 4 - 12 + c
    so, c = 7

    y = x^2 - 6x + 7

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