Tangent Line of f(x) = x - 2x^3 + x^4

shows that the the tangent line at coordinate , given by for the example exactly captures the notion of a line being tangent to a curve.
Some questions to think about: (1) When reaches a local minimum point, what is the slope of the tangent line? How about a local maximum point? (2) is knowing the slope of a tangent line at enough to tell if is maximal or minimal? (3) is it possible for the tangent line to be vertical?