calculus

6. Determine a, b, c, and d so that the graph of y=ax^3+bx^2+cx+d has a point of inflection at the origin and a relative maximum at the point (2, 4). Sketch the graph.

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  1. y = ax^3 + bx^2 + cx + d
    y' = 3ax^2 + 2bx + c
    y'' = 6ax + 2b

    y''(0) = 0
    so b = 0

    y'(2) = 0
    so c = -12a

    y(0) = 0
    so d = 0

    y = ax^3 - 12ax
    y(2) = 4
    so a = -1/4

    y = -1/4 x^3 + 3x
    y' = -3/4 x^2 + 3
    y'' = -3/2 x

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