A cubic curve is given by an equation of the form

y = a x^{3} + b x^{2} + c x + d, a \neq 0

It looks a bit like a quadratic, but there is a new term in $x^{3} .$ This new term will change the shape of the graph we get, as you will see in the next examples.

Three cubics are defined as

\begin{aligned} f (x) & = (x - 3) (x - 2) (x + 1) \\ g (x) & = (x + 4) (x^{2} + x - 2) \\ h (x) & = (2 - x) (x^{2} + 10 x + 25) \end{aligned}

Sketch the graphs of these functions, indicating the intersections with the axes. You do not need to calculate the turning points.

Problem

Find which curve below corresponds to

y = (x + 4) (x^{2} + 2 x + 2)

and explain your answer.

Problem

A sphere and a cone are constructed, where the radius of the sphere is equal to the radius of the circular base of the cone. The height of the cone is two units larger than the radius of its base.

Given that the sphere and the cone have the same volume, find the value of the radius shared by the sphere and the cone.

Problem

You are given that any cubic must have at least one real root (the proof of this is quite hard, we assume it for now).

Given any quadratic and any cubic, prove that they must eventually intersect.

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