Chapter 4 | Applications of Derivatives
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f (0) =1. It occurs at x =0, as shown in Figure 4.13 (b). A function may have both an absolute maximum and an absolute minimum, just one extremum, or neither. Figure 4.13 shows several functions and some of the different possibilities regarding absolute extrema. However, the following theorem, called the Extreme Value Theorem , guarantees that a continuous function f over a closed, bounded interval [ a , b ] has both an absolute maximum and an absolute minimum.
Figure 4.13 Graphs (a), (b), and (c) show several possibilities for absolute extrema for functions with a domain of (−∞, ∞). Graphs (d), (e), and (f) show several possibilities for absolute extrema for functions with a domain that is a bounded interval.
Theorem 4.1: Extreme Value Theorem If f is a continuous function over the closed, bounded interval [ a , b ], then there is a point in [ a , b ] atwhich f has an absolute maximum over [ a , b ] and there is a point in [ a , b ] at which f has an absolute minimum over [ a , b ].
The proof of the extreme value theorem is beyond the scope of this text. Typically, it is proved in a course on real analysis. There are a couple of key points to note about the statement of this theorem. For the extreme value theorem to apply, the
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