10
extrema
Types, formula usage, and applications
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Extrema
Definition of an Extrema

The extrema of a function f are the values where f is either a maximum or a minimum. More rigorously, we have Let f be a function defined on the interval (a,b) containing the point c. Then * f has minimum at c if f(c) < f(x) for all x in (a,b). * f has maximum at c if f(c) > f(x) for all x in (a,b). The following definition gives the types of minimums and/or maximums values Definitions:

1. We say that f(x) has an absolute (or global) maximum at x=c if f(c) > f(x) for every x in the domain we are working on. 2. We say that f(x) has a relative (or local) maximum at x=c if f(c) > f(x) for every x in some open interval around x=c 3. We say that f(x) has an absolute (or global) minimum at x=c if f(c) < f(x) for every x in the domain we are working on. 4. We say that f(x) has a relative (or local) minimum at x=c x=cx=x if f(c) < f(x) for every x in some open interval around x=c. Note that when we say an “open interval around x=c ” we mean that we can find some interval (a, b), not including the endpoints, such that a<c<b. Or, in other words, c will be contained somewhere inside the interval and will not be either of the endpoints. Also, we will collectively call the minimum and maximum points of a function the extrema of the function. So, relative extrema will refer to the relative minimums and maximums while absolute extrema refer to the absolute minimums and maximums. Now, let’s talk a little bit about the subtle difference between the absolute and relative in the definition above. We will have an absolute maximum (or minimum) at x=c provided f(c) is the largest (or smallest) value that the function will ever take on the domain that we are working on. Also, when we say the “domain we are working on” this simply means the range of x’s that we have chosen...