41 - Absolute and local minimum and maximum values Defn: A...

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Unformatted text preview: Absolute and local minimum and maximum values Defn: A function has an absolute maximum at if domain of . The number is called the maximum value of A function has an absolute minimum at if . The number is called the minimum value of The maximum and minimum values of on . for all on . in in , where is the for all , where is the domain of are called the extreme values of . for all near . con tainin g such that for all , then near . and an absolute for all in A function has a local maximum at if In more formal language, if there is an open interval . A function has a local minimum at if The Extreme Value Theorem: If is continuous on a closed interval minimum at some numbers and in attains a maximum value min and max at end points min and max in interior points where derivative doesn't exist MTH 173 section 4.1 notes, page 1 mn and max at interior points where derivative is min at interior point where derivative is , max at end point min at interior point where derivative doesn't exist, max at end point Fermat's Theorem: If has a local maximum or minimum at and if exists, then The essense of this theorem is that the only possible places that local max or min can occur is where the derivative is or where the derivative fails to exist. Thus candidates for local extrema are found by determining where the derivative is or doesn't exit. We call these numbers critical numbers. MTH 173 section 4.1 notes, page 2 Defn: A critical number of a function or does not exist. Proof of Fermat's thm: is a number in the domain of such that either Assume has a local maximum at . Then we know that close eno gh to , we know that for all near . Thus for If , then and then lim so If , then so lim so now we have that both and . The only way that can happen is for MTH 173 section 4.1 notes, page 3 determine critical numbers of 32. always exits, so critical numbrs occur where 38. the domain excludes for . This is the only critical number. MTH 173 section 4.1 notes, page 4 when . fails to exist when the critical numbers are determine abs max and min of fn in interval 48. Here we determine interior critical numbers, then evaluate the function at these critical numbers and at the end points, and finally chooase the smallest and largest function values for these choices: is a critical number in the minimum is and the maximum is MTH 173 section 4.1 notes, page 5 critical numbers in are the minimum in , the maximum is ! when the minimum is and the maximum is MTH 173 section 4.1 notes, page 6 ! " " " when " ln " " ln " " " " " " " " ln " " # ! " " ln " " # the minimum is and the maximum is MTH 173 section 4.1 notes, page 7 ...
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