# Mutual tangent lines

Find all points where the curves $f(x) = x^3-3x+4$ and $g(x) = 3x^2-3x$ share the same tangent line.

Graphing them I see that they look like they share a tangent line at $x=2$.

I got the derivatives of both and set them equal to each other and got $x=0$ and $x=2$.

After plugging $2$ back in I got $6$. So the point is $(2,6)$.

Is that correct?

• In hindsight, this is not a duplicate. It appeared to be a duplicate because the OP originally had a scan of a problem sheet that included this problem, as well as the one from the other question. I have altered the other question to make this clear. – Cameron Buie Nov 2 '13 at 13:15

$$f' = g ' \iff 3x^2 - 3 = 6x - 3 \iff x^2 -2x = 0 \iff x(x-2) = 0 \iff x=0,2$$
$$(2, 6 )$$ as you said. Notice at $0$ functions are not equal, so they cannot share a tangent line at $0$