Lesson 4.1 Extreme Values

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  • 1. 4.1 Extreme Values of Functions
  • 2. 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.
  • 3. 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
  • 4. 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?
  • 5. 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
  • 6. 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
  • 7. 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.
  • 8. 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
  • 9. 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.
  • 10. 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.