Can a cubic function have two tangents at a single point?

Because sometimes a picture really is worth a thousand words:

but I can't see how there could be two distinct tangents at a given point.

Very easy: this point does not belong to the curve. There exist exactly two tangents to the curve, touching the curve at different points, that intersect at this point.

It's just a case of misreading: Take the unit circle and the point $P = (3, 0)$. You can draw "two tangents from $P$ to the unit circle."

The same deal here: you've got the graph of a cubic, and if you take the right two tangents, they MEET at the point $(h, 3-h)$. I'm almost certain this is what the question is asking.