121616949-math.121 - 6.1 Optimization 107 local maximum...

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6.1 Optimization 107 local maximum must be the global maximum and the smallest local minimum must be the global minimum. It is usually easier, however, to compute the value of f at every point at which the global maximum or minimum might occur; the largest of these is the global maximum, the smallest is the global minimum. So we compute f ( - 2) = 4, f (0) = 0, f (1) = 1. The global maximum is 4 at x = - 2 and the global minimum is 0 at x = 0. It is possible that there is no global maximum or minimum. It is difficult, and not particularly useful, to express a set procedure for determining whether this is the case. Generally, the best approach is to gain enough understanding of the shape of the graph to decide. Fortunately, only a rough idea of the shape is usually needed. There are some particularly nice cases that are easy. If you have a continuous function on a closed interval [ a, b ], there is always both a global maximum and a global minimum, so examining the critical values and the endpoints is enough: THEOREM 6.2
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