Q.a.Find the stationary points on the graph of y=12x+3x^2-2x^3 and sketch the curve.
b.How does your sketch show that the equation 12x+3x^2-2x^3 =0 has exactly three real roots.
c.Use your graph to show that the eequation 12x+3x^2-2x^3 = -5 also has exactly three real roots.
d.For what range of values of k does the equation 12x+3x^2-2x^3 = k have i)exactly three real roots ii)only one real root?

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  1. a) y=12x+3x^2-2x^3
    y' = 12+6x-6x^2 = 6(2-x)(1+x)

    b) y(-3) > 0 and y(-1) < 0 and y(0) = 0 and y(1) > 0 and y(4) < 0
    c) the line y = -5 also crosses the graph at three places

    d) i) If the two stationary points are a min at y=a and a max at y=b, then a < k < b
    ii) k < a or k > b

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  2. Can you write the answer of only d)

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