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

shows that the the tangent line

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?

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