derivatives

For their points of intersection:

3x2 = 2x3 + 1

2x3 - 3x2 + 1 = 0

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

x = 1 or -1/2

For their derivatievs:

For y = 3x2, y' = 6x and so y' = 6 and 3 at x = 1 and -1/2 resp.

For y = 2x3 + 1, y' = 6x2 and so y' = 6 and 3/2 at x = 1 and -1/2 resp.

So the common tangent is at x = 1, y = 3, with equation:

(y - 3)/(x - 1) = 1

y - 3 = x - 1

x - y + 2 = 0