Basically, the two curves are required to have the same gradient at their point of intersection.
If you were to do this algebraically:
Let f(x) = x³ - x + 2 and g(x) = mx
For POI: x³ - (m + 1)x + 2 = 0 ...(1)
Also, f'(x) = g'(x) -> 3x² - 1 = m ...(2)
Solve simulateously. The POI turns out to be (1, 2) and m = 2.
I'm not really sure how to verbally express an argument solely based on graphical properties. Anyone else?