for what range of values of k does the equation 12x+3x^2-2x^3=k have  a) Exactly three real roots? b) Two real roots? c) Only one real root?

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We need to find where the turning points are of the cubic on the lefthand side of the equation. We do this by differentiating the cubic with respect to `x`:

`d/(dx) (12x + 3x^2 - 2x^3) = 12 + 6x - 6x^2`

To find the turning points we solve

`-6x^2 + 6x + 12 = 0`

ie, `x^2 - x - 2 = 0`

By insepction

`(x+1)(x-2) = 0`

So the turning points are at `x = -1` and `x = 2`

When `k` is between...

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