Maximisation and Minimisation
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In the last chapter, we were concerned with the effect a (small) change in one variable had on another. For example, we investigated how a small change in the quantity produced, influenced the Total Cost of a firm. In this chapter, we centre our attention on the shape of a function; in particular, we are looking for the ‘hills’ and ‘valleys’ of the function. Frequently, the decision-maker (e.g. household, entrepreneur) wants to decide upon a course of action that will maximise (i.e. find a ‘hill’) or minimise (i.e. find a ‘valley’) of something. In mathematical terms, we say we are searching for the extrema. Depending upon the function, there can be several maximum and minimum points (i.e. relative or local extrema). The optimal (‘best’) result, which we call the global extremum, has to be found by comparison of the relative extrema and the end points of the function. To find such extrema, it is usually sufficient to refer to the first and second derivatives.
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