The tangent line to a curve at a point on the curve can never cross the curve at the point of tangency. Tor F?

Answer 1

False.

Some graphs that have point(s) of inflection may have tangent lines at those point(s) of inflection where the graph cuts across the tangent line.

[a necessary condition for a point of inflection :
if (c, f(c)) is a point of inflection of the graph of the function f, then either f'' (c) = 0 or f'' (c) is not defined.]

typical example from a book will show a concave downward portion of the graph of f , a point of inflection and then a concave upward portion (think sine or cosine wave )

check any calc book - somewhere in the vicinity of concavity and 2nd-deriv tests in the book they will talk about inflection points and you'll see suitable examples

Answer 2

it does not cut the curve at the point of tangency, this is right, it is just tangent to it,
but the line (at a different point) might cross the curve again.