shows that the the tangent line [math]g[/math] at [math]x[/math] coordinate [math]m[/math], given by [math]g(x) = f(m)+f'(m)(x-m)[/math] for the example [math]f(x) =x - 2x^3 + x^4[/math] exactly captures the notion of a line being tangent to a curve.
Some questions to think about: (1) When [math]m[/math] 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 [math]m[/math] enough to tell if [math]m[/math] is maximal or minimal? (3) is it possible for the tangent line to be vertical?