Consider the cubic curve

Find values for and such that the line is tangent to the graph of at the point .

There is a second line passing through the point which is tangent to the graph of at the point . Find the values of and .

First, we sketch the graph of on the interval :

We compute the derivative of ,

So, the slope of the tangent line at the point is

Then, since is the tangent line to the curve at it must be on the curve at this point, so

Therefore, the line is , i.e., .

Next, if there is another line, say , tangent to at a point we know it has slope given by

Since it must pass through the point (by hypothesis) we must have

So the line is of the form,

Finally, since the point is on both this line and the curve we have,

Since these are both equal to , they must be equal to each other,

Since we then have .

Thus, , and are the requested values.