Consider the curve given by the cubic equation

Show that the line is tangent to this curve and find the point of tangency. Determine if this line intersects the curve anywhere else.

First, we compute the derivative of the curve,

For the line to be tangent to the curve it must be at a point such that (since this line has slope , the derivative of the curve must be ). So,

Next, for the line to be tangent to the curve the point must actually be on the curve. So, we test out the two possible values of .

If , we have . So is not tangent to the curve at .

If , we have ; hence, is tangent to the curve at .

This tangent line also intersects the curve at .