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Stitz-Zeager_College_Algebra_e-book

# The point 4 3 fails for the same reason no open

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Unformatted text preview: e, we say the maximum11 of f is 5.5; similarly, the minimum12 of f is −8. We formalize these concepts in the following deﬁnitions. Definition 1.9. Suppose f is a function with f (a) = b. • We say f has a local maximum at the point (a, b) if and only if there is an open interval I containing a for which f (a) ≥ f (x) for all x in I . The value f (a) = b is called ‘a local maximum value of f ’ in this case. • We say f has a local minimum at the point (a, b) if and only if there is an open interval I containing a for which f (a) ≤ f (x) for all x in I . The value f (a) = b is called ‘a local minimum value of f ’ in this case. • The value b is called the maximum of f if b ≥ f (x) for all x in the domain of f . • The value b is called the minimum of f if b ≤ f (x) for all x in the domain of f . It’s important to note that not every function will have all of these features. Indeed, it is possible to have a function with no local or absolute extrema at all! (Any ideas of what such a function’s graph would have to look like?) We shall see in the exercises examples of functions which have one or two, but not all, of these features, some that have instances of each type of extremum and...
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