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.