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Lesson 4.1 Extreme Values
 

Lesson 4.1 Extreme Values

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    Lesson 4.1 Extreme Values Lesson 4.1 Extreme Values Presentation Transcript

    • 4.1 Extreme Values of Functions
    • Def : Absolute Extreme Values
      • Let f be a function with domain D. Then f (c) is the…
      • a) Absolute (or global) maximum value on D iff f (x) ≤ f (c) for all x in D.
      • b) Absolute (or global) minimum value on D iff f (x) ≥ f (c) for all x in D.
      • * Note: The max or min VALUE refers to the y value.
    • Where can absolute extrema occur? * Extrema occur at endpoints or interior points in an interval. * A function does not have to have a max or min on an interval. * Continuity affects the existence of extrema. Min Max Min No Max No Max No Min
    • Thm 1 : Extreme Value Theorem
      • If f is continuous on a closed interval [a,b], then f has both a maximum value and a minimum value on the interval.
      * What is the “worst case scenario,” and what happens then?
    • Def : Local Extreme Values
      • Let c be an interior point of the domain of f . Then f (c) is a…
      • a) Local (or relative) maximum value at c iff f (x) ≤ f (c) for all x in some open interval containing c.
      • b) Local (or relative) minimum value at c iff f (x) ≥ f (c) for all x in some open interval containing c.
      * A local extreme value is a max or min in a “neighborhood” of points surrounding c
    • Identify Absolute and Local Extrema * Local extremes are where f ’ (x)=0 or f ’ (x) DNE Abs & Loc Min Abs & Loc Max Loc Max Loc Min Loc Min
    • Thm 2 : Local Extreme Values
      • If a function f has a local maximum value or a local minimum value at an interior point c of its domain, and if f ’ exists at c , then f ’(c)=0.
    • Def : Critical Point
      • A critical point is a point in the interior of the domain of f at which either…
          • 1) f ’(x)= 0 , or
          • 2) f ’(x) does not exist
    • Finding Absolute Extrema on [a,b]…
      • To find the absolute extrema of a continuous function f on [a,b]…
      • Evaluate f at the endpoints a and b.
      • Find the critical numbers of f on [a,b].
      • Evaluate f at each critical number.
      • Compare values. The greatest is the absolute max and the least is the absolute min.
    • Check for Understanding… (True or False??)
      • Absolute extrema can occur at either interior points or endpoints.
      • A function may fail to have a max or min value.
      • A continuous function on a closed interval must have an absolute max or min.
      • An absolute extremum is also a local extremum.
      • Not every critical point or end point signals the presence of an extreme value.