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a vetical line has an undifined rate of change
A minimum value (of any function, not just a polynomial) is a value that has a lower value than any nearby value. A global minimum is a value that is lower than ANY other value. (This answer is just a brief and informal overview; check the Wikipedia article on "maxima and minima" for a more detailed explanation.)
An "extreme value" is either a local maximum, or a local minimum - i.e., a point which is greater than all the points in a certain neighborhood, or less than all points in a certain neighborhood.