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There are two tests involved in checking for maximum and minimum because, if you only checked for a value of zero for the first derivative, you would only know that the equation has zero slope at that point. You also need to check the second derivative to see if that point is a maximum, a minimum, or an inflection point.

To be fully correct, you also need to understand the equation itself, because there may be more than one maxima or minima, and/or there may be a discontinuity. This is all part of the process of finding a maximum or minimum.

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